# ConSep: a Noise- and Reverberation-Robust Speech Separation Framework by
Magnitude Conditioning

Kuan-Hsun Ho  Department of Computer Science and
Information Engineering
National Taiwan Normal University
Taipei, Taiwan
Email<EMAIL_ADDRESS>Jeih-weih Hung  Department of Electrical
Engineering
National Chi Nan University
Nantou, Taiwan
Email<EMAIL_ADDRESS>Berlin Chen  Department of Computer Science and
Information Engineering
National Taiwan Normal University
Taipei, Taiwan
Email<EMAIL_ADDRESS>
###### Abstract

Speech separation has recently made significant progress thanks to the fine-
grained vision used in time-domain methods. However, several studies have
shown that adopting Short-Time Fourier Transform (STFT) for feature extraction
could be beneficial when encountering harsher conditions, such as noise or
reverberation. Therefore, we propose a magnitude-conditioned time-domain
framework, ConSep, to inherit the beneficial characteristics. The experiment
shows that ConSep promotes performance in anechoic, noisy, and reverberant
settings compared to two celebrated methods, SepFormer and Bi-Sep.
Furthermore, we visualize the components of ConSep to strengthen the
advantages and cohere with the actualities we have found in preliminary
studies.

###### Index Terms:

speech separation, reverberation, cross-domain, multi-resolution, magnitude,
conditioning

##  I Introduction

SPEECH separation is a specific scenario of the source separation problem,
where the targets are the overlapping speech signal sources while other
irrelevant signals are treated as interferences. Recently, the field of speech
separation has been revolutionized with the advent of deep learning
techniques. The previous works toward an anechoic environment [1, 2, 3, 4, 5,
6, 7] have been fruitful and inspiring. Current systems rely, in large part,
on the prestigious EMD structure, which is composed of the Encoder, Mask
estimator, and Decoder. However, the assumption of an anechoic environment
might be unrealistic in most speech separation studies. In practice, speech
usually coincides with various interferences, and speech separation under
reverberant situations is incredibly challenging [8].

Interestingly, we have observed opposing performances in [9] and [10]. In [9],
the experimental results show that a smaller-sized SepFormer (SepFormer-s),
i.e., the SepFormer with fewer Inter- and Intra-Transformers, could not
perform well under reverberant conditions despite still giving a competitive
result under an anechoic condition. After optimizations toward reverberant
conditions, the resulting SepFormer is marginally superior to the optimized
PIT-BLSTM. Moreover, it fails to be competitive in an anechoic condition. (See
[9, Tab. 6]) Meanwhile, [10] expands the work in [7] on a regular-sized
SepFormer and claims good results on multiple conditions. One explanation may
be the advantage of exploiting more parameters, which enables the model to
represent more complicated functions than the one with fewer parameters.
Nonetheless, this evidences that efforts to make SepFormer a more distilled
yet versatile model need further investigation.

One endeavor that adapts SepFormer to multiple conditions is Bi-Sep [11]. Bi-
Sep leverages two parallel encoders with different time resolutions and a Bi-
projection Fusion (BPF) [12] module to integrate information from different
domains. However, Bi-Sep has a potential shortcoming. Despite exhibiting
better performance than existing models when facing complicated environments,
Bi-Sep inevitably inherits the degradation when substituting the learnable
encoder with the Short-Time Fourier Transform (STFT). Moreover, although the
BPF module helps determine whether the mask estimator should attend more on a
shorter or longer frame, it could cause the mask estimator to have difficulty
learning from two domains when encountering a simpler environment.

Therefore, we propose a novel speech separation framework, ConSep, which
exploits conditioning on magnitude spectrogram to avoid any domain mismatch or
confusion. A better conditioning method could facilitate the knowledge
injection instead of concatenating all the features as Bi-Sep does. ConSep
likewise embraces two encoders with different time resolutions to retrieve
complementary characteristics. However, we merge the respective output
features into the same dominant domain, in this case, the time domain. More
precisely, we modulate the time signals by magnitude spectrogram as this
modulation enables the mask estimator to better distinguish speech parts and
interferences during separation. Experiments show that ConSep surpasses
SepFormer under an anechoic condition and prominently upgrades SepFormer under
more complicated situations. This result matches our goal of enabling a model
to possess the generalizability on multiple conditions.

##  II Findings

TABLE I: Best model configuration from references that adapt EMD models to
multiple conditions and the original SepFormer.

Condition(s) Attributes Anechoic Loss function Granularity (ms)
Encoder/Decoder pair SepFormer[7] SI-SDR 2 Learnable Conv-TasNet[16] SI-SDR
0.5 Learnable3 Reverberant Loss function Granularity (ms) Encoder/Decoder pair
SepFormer-s[9] th-SDR 64 STFT4 SepFormer[10] SI-SDR 2 Learnable Conv-
TasNet[16] SI-SDR 8 STFT3 DenseUNet-TCN[18] L11 25 STFT WD-TCN [13] SI-SDR 2
Learnable Bi-Sep32 [11] SI-SDR 2/322 Both 1 L1 Loss on real, imaginary and
magnitude. 2 Frame length of 2 and 32 ms for Learnable and STFT, respectively.
3 Performance reduces by less than 2 (dB) if altered to the other pair. 4
Performance reduces by less than 1 (dB) if altered to the other pair.

To ascertain what fosters a model to cope with various environments, we list
the optimal model configurations of relevant studies that adapt EMD models to
more complicated conditions in Tab. I. We notice that an almost unanimous
consensus is to employ the time-domain loss, including SI-SDR [14] and th-SDR
[15]. The time-domain loss has been proven beneficial in numerous works, even
when using STFT as an encoder [9, 16]. However, the granularity designated by
each work varies over a wide range, from 0.5 ms to 64 ms. The time-domain
methods usually work on short frames, whereas traditional STFT frame sizes are
set to be larger (around 32 ms). As for the type of encoder-decoder pair, [16]
and [17] have argued that it is not the crucial factor to success.

Furthermore, through our preliminary studies, we discover four actualities:

  * 1)

Time-domain methods usually perform better in SI-SDR and worse in PESQ than
STFT methods [18, 19, 20].

  * 2)

A large enough window size is mandatory to avoid contravening the prerequisite
of Multiplicative Transfer Function Approximation (MTFA) [9, 21]. On the
contrary, recent works have gained success due to the fine-grained window size
[16].

  * 3)

Employing STFT representation exhibits optimal performance in reverberation.
However, employing the learnable encoder/decoder prevails under an anechoic
condition [9, 16].

  * 4)

The phase becomes uninformative within a relatively large window size [22, 23,
24].

Although some actualities seem contradictory, a reasonable scheme to
incorporate all the beneficial characteristics can compensate for those
contradictions. This motivates us to propose ConSep. To encourage more
instantiations in the future, we plan to provide publicly available codes used
in our experiments.

##  III Our ConSep

The high-level description of ConSep is identical to the EMD structure.
Initially, the encoder transforms the mixture $x\in\mathbb{R}^{T}$, which
contains audio from $K$ active speakers and interferences, into a
representation $w$ that characterizes the signal. Then the mask estimator
produces $K$ masks $\\{m_{k}\\}$ for each active speaker in the mixture.
Finally, the decoder reconstructs the separated $K$ source signals in the time
domain, each represented by $\hat{s}_{k}\in\mathbb{R}^{T}$.

###  III-A Encoder

The encoder of ConSep is composed of four modules: Learnable encoder, STFT,
Multi-Channel Attention (MulCA), and Modulator.

####  III-A1 Learnable encoder

The typical method uses one-dimensional convolutional layers (Conv1d) followed
by a rectified linear unit (ReLU). This encoder extracts the time
representation $w_{c}\in\mathbb{R}^{N\times L}$ from the mixture $x$:

$w_{c}=\text{ReLU}(\text{Conv1d}(x)).$ (1)

####  III-A2 STFT with MulCA

The conventional method obtains the time-frequency representation
$X\in\mathbb{C}^{F\times L}$, a.k.a spectrogram, through STFT:

$X[f,l]=\sum_{n=0}^{W-1}x[n+Hl]w[n]e^{-j\frac{2\pi fn}{W}},$ (2)

where $f$, $l$, and $n$ are the frequency bin, frame, and local time indices;
$W$ is the window length, and $H$ is the hop size. After, we extract the
magnitude part $X_{m}\in\mathbb{R}^{F\times L}$ from the spectrogram $X$.
Prior to the modulation, we add a MulCA block [26, 25] to weigh the magnitude
spectrogram $X_{m}$. MulCA regards different frequency bins as channels,
giving them different weights using Channel Attention.

The intuition behind MulCA is that the energy of a speech utterance usually
distributes non-uniformly in frequencies, and different frequency components
are unequally crucial to human perception [27, 28]. For example, the lower
frequency band tends to contain high energies, tonalities, and long-duration
sounds; the higher frequency band may have low energies, noise, and rapidly
decaying sounds. The following equations express the operations of a MulCA:

$\begin{gathered}c_{i}=\text{ReLU}(\text{AvgPool}(\text{Conv1d}(X_{m};k_{i}))),i=0,1,2,\\\
c=\text{FCN}([c_{0},c_{1},c_{2}]),c\in\mathbb{R}^{F},\\\
C=\text{Broadcast}(c),C\in\mathbb{R}^{F\times L},\\\ \tilde{X}_{m}=X_{m}\odot
C.\end{gathered}$ (3)

The frames in $X_{m}$ are passed through three Conv1d with different kernel
sizes: $k_{0}$(small), $k_{1}$(middle), and $k_{2}$(large). Each is followed
by an average pooling (AvgPool) and ReLU activation to deliver a weight vector
$c_{i}$. Afterward, a two-layer down-up-sampled fully connected network (FCN)
merges three weight vectors to create frequency-wise weights $c$. Then we
broadcast the weights to operate element-wise multiplication with $X_{m}$ to
get a weighted version $\tilde{X}_{m}$—accordingly, all the frames in the
magnitude spectrogram share an identical frequency-dependent emphasis.

####  III-A3 Modulator

As mentioned earlier, we aim to explore a better conditioning method to remain
in the same domain without encountering any domain conflicts. The mask
estimator of SepFormer has shown its powerful ability to model sources from
overcomplete features in the time domain. Hence, we build ConSep accordingly,
viz., analyzing time-domain features, but additionally, the time signals are
conditioned on the magnitude spectrogram. We rely on Feature-wise Linear
Modulation (FiLM) [29]. The FiLM allows adjusting the time signals to more
appropriate representations based on the energy information of the given
spectrogram. The residual connection is added after the FiLM layer to ensure
the architecture performs well when the magnitude is relatively small. The
modulating process can be formulated as:

$w=w_{c}+f_{1}(\tilde{X}_{m})\odot w_{c}+f_{2}(\tilde{X}_{m}),\\\ $ (4)

where $w$ denotes the modulated feature, and each $f_{i}$ denotes an affine
transformation.

###  III-B Mask estimator

The mask estimator inputs modulated feature $w$ and estimates a mask $m_{k}$.
Firstly, the modulated feature $w$ is layer-wise normalized, chunked into
overlapping segments with an overlap factor of 50%, and then stacked.

Afterward, the stacked feature feeds the SepFormer blocks, which exploit the
dual-path mechanism [4]. The underlying process first captures the short-term
dependencies by Intra-Transformer, then extracts the long-term dependencies by
Inter-Transformer, and repeats $D$ times. The unit Transformer structure used
in Intra- and Inter-Transformer includes a multi-head attention (MHA) stage
and a feed-forward (FFW) block with pre-LN setting [30] and skip connections.
Unlike DPTNet [6], positional encoding is applied for injecting information on
the order of sequence instead of a recurrent neural network (RNN). The total
number of unit Transformers employed in Intra- and Inter-Transformer is $E$. A
linear layer further processes the output of the SepFormer block to project
the feature dimension for $K$ times deep.

Finally, the projected output is passed through an overlap-add stage, two
concurrent FFWs, and a ReLU activation to obtain the non-negative mask
$m_{k}$. Note that we eliminate the bottleneck projections in the original
SepFormer, as our prior experiment indicates its redundancy.

###  III-C Decoder

The decoder is a transposed convolution layer with the same stride and kernel
size as the learnable encoder. The input to the decoder for active speaker $k$
is the element-wise multiplication between the mask $m_{k}$ and the modulated
feature $w$:

$\hat{s}_{k}=\text{Conv1d}^{T}(m_{k}\odot w).$ (5)

##  IV Experimental setup

###  IV-A Datasets

We validate the presented method on the popular WSJ0-2mix dataset [1] for the
anechoic setting using the improvement of SI-SDR and SDR as the evaluation
metrics. The training, validation, and test sets contain 30, 10, and 5 hours
of speech data. The speech data are sampled at 8 kHz. Furthermore, we perform
experiments in noisy settings. We rely on WHAM! [31] with urban noise and
WHAMR! [8], which adds reverberation on top of WHAM!. These datasets are
derived from WSJ0-2mix and have identical statistics.

###  IV-B Training setup

In the case of the learnable encoder, the encoder basis $N$ is set to 256. The
input kernel size is 16 with a stride factor of 8. As for the STFT encoder, we
use the Hamming window with a length of 256 (32 ms at 8 kHz), and the hop size
is the same as that for the learnable encoder. For the MulCA, three kernel
sizes $k_{0}$, $k_{1}$, and $k_{2}$ are 3, 5, and 10, respectively. Regarding
the mask estimator, we follow the configuration proposed in [7], whereas $E$
is reduced to 4. For model training, we optimize the model using the Adam
optimizer, a batch size of 1, and a learning rate of 1.5e-4. Finally, the
model is trained over 150 epochs with utterance-level Permutation Invariant
Training (uPIT) [2] and SI-SDR losses.

##  V Results and analyses

###  V-A Comparison and Ablation study

TABLE II: Separation performance and speech quality metric on multiple
conditions.

Condition(s) | Metrics
---|---
Anechoic |  SI-SDRi |  SDRi |  NB-PESQ
SepFormer-s | 16.53 | 17.02 | 3.12
Bi-Sep32 | 16.28 | 16.49 | -
ConSep | 16.72 | 17.19 | 3.39
Noisy | SI-SDRi | SDRi | NB-PESQ
SepFormer-s | 13.23 | 14.13 | 2.50
Bi-Sep32 | 13.62 | 14.54 | -
ConSep | 13.82 | 14.82 | 2.76
Noisy & Reverberant | SI-SDRi | SDRi | NB-PESQ
SepFormer-s | 5.90 | 8.95 | 2.12
Bi-Sep32 | 6.37 | 9.09 | -
ConSep | 6.50 | 9.07 | 2.30

TABLE III: Ablation study. The third and fourth columns use different
conditioning methods without MulCA as well.

| ConSep | w/o MulCA |  | w/o FiLM
---
(concat+linear)
| w/o FiLM
---
(add)
| w/o
---
conditioning
SI-SDRi | 13.82 | 13.72 | 13.59 | 13.34 | 13.23
SDRi | 14.82 | 14.68 | 13.96 | 14.19 | 14.13

We compare the separation accuracy of ConSep with SepFormer-s and Bi-Sep as
baselines. Tab. II presents the experimental results in terms of both SI-SDRi
and SDRi. It demonstrates the advantages of our proposed model. For all kinds
of environments, ConSep outperforms all other methods except the SDRi, which
can be deceived by the loudness [14], in noisy and reverberant settings.
Moreover, the fact that a better conditioning strategy not only gains
stability but succeeds to the beneficial sides is proven, as ConSep surpasses
Bi-Sep in non-anechoic settings and SepFormer in the anechoic setting.

Furthermore, we show the evaluation in speech quality metric. We can observe
that employing the STFT features may improve narrow-band PESQ (NB-PESQ), and
the same phenomenon can be observed in [18, 19, 20]. This adds another clue of
using a larger frame time-frequency representation still presenting as a
desirable feature, even if learned-domain models have more freedom to adapt to
the SI-SDR training loss.

To validate the effectiveness of ConSep, we also conduct an ablation study in
noisy environments, as shown in Tab. III. We validate the performance without
MulCA and magnitude conditioning. Additionally, we experiment with various
conditioning methods, including simply adding or concatenation followed by a
down-sizing linear layer. The result shows that with attention to magnitudes,
the separation performance improves, as ”concat+linear” performs closely to
Bi-Sep. However, employing FiLM brings the most apparent improvement among
other methods. (see the entry ”w/o MulCA”)

###  V-B Visualization

(a) Anechoic

(b) Noisy

(c) Noisy & Reverberant

Figure 1: Case studies. Generally, the rows indicate the spectrogram of
mixture, sources, ConSep output, and SepFormer output from top to bottom. The
two columns indicate the first and second sources from left to right. Also,
red and blue boxes denote false alarm issues and spectral/harmony clarity. For
(a) and (b), the non-speech signals cropped in the orange box are the sounds
of inhaling and microphone pop, respectively.

(a) ConSep

(b) SepFormer-s

Figure 2: For each sub-figure, the upper and lower panel depict the encoder
bases sorted by Euclidean similarity and their frequency response,
respectively.

We visualize the mixture, clean sources, and separated outputs from ConSep and
SepFormer, as shown in Fig. 1. The mixture in Fig. 1(a) consists of two women
speaking with a similar pitch range. We can see that the frequency contour is
more prominent in ConSep-separated sources and that SepFormer tends to produce
false alarms. This implies that merely analyzing time signals from an
overcomplete set of encoder bases renders the model confusing when facing
speakers with similar pitch identities. As for Fig. 1(b), the mixture consists
of a woman, a man speaking, and cafeteria noises. We can see that both models
denoise well, but likewise, some speech-related false alarms occur. The same
observation can also be pointed out in Fig. 1(c), whose mixture consists of
the reverberant version of two men speaking with poll hall noises. Regardless,
the superiority of a more apparent contour in ConSep remains, probably due to
the enhancement by attending the spectrogram that better presents the
harmonics. This concludes that the essential components of speech signals are
better captured in ConSep.

Furthermore, we plot the sorted bases of the learnable encoder trained in
anechoic condition and their frequency response, as shown in Fig. 2.
Resembling [3], most filters are tuned to lower frequencies. This suggests an
essential role for low-frequency speech features such as pitch to achieve
better performance. However, we notice a few weird filters on the right side
of Fig. 2(b), which can translate to the high-valued low-pass filter at the
end of the frequency response. This may be the reason why SepFormer could not
perform well when attention to high-frequency information is required, such as
the circumstances when facing similar-pitch speakers.

##  VI Conclusions

In this study, we propose a noise- and reverberation-robust speech separation
framework, ConSep, by means of conditioning the time signals by magnitude
spectrogram. The goal of generalizability has been fulfilled as this framework
upgrades an existing model to fit various environments. Furthermore, we
analyze and visualize the results to get a better picture of the advantages of
ConSep, which as well demonstrates phenomena coherent with the actualities
found through preliminary studies.

## References

  * [1] Y. Isik, J. L. Roux, Z. Chen, S. Watanabe, and J. R. Hershey, “Single-channel multi-speaker separation using deep clustering,” in INTERSPEECH, 2016.
  * [2] M.Kolbæk, D.Yu, Z.Tan, and J.Jensen, “Multitalker speech separation with utterance-level permutation invariant training of deep recurrent neural networks,” IEEE/ACM Trans. Audio Speech Lang. Process., vol. 25, no. 10, pp. 1901–1913, 2017.
  * [3] Y. Luo and N. Mesgarani, “Conv-TasNet: Surpassing Ideal Time–Frequency Magnitude Masking for Speech Separation,” IEEE/ACM Trans. Audio Speech Lang. Process., vol. 27, no. 8, pp. 1256–1266, Aug. 2019.
  * [4] Y. Luo, Z. Chen, and T. Yoshioka, “Dual-Path RNN: Efficient Long Sequence Modeling for Time-Domain Single-Channel Speech Separation,” in ICASSP, 2020.
  * [5] N. Zeghidour and D. Grangier, “Wavesplit: End-to-End speech separation by speaker clustering,” IEEE/ACM Trans. Audio Speech Lang. Process., vol. 29, pp. 2840–2849, 2021.
  * [6] J. Chen, Q. Mao, and D. Liu, “Dual-Path Transformer Network: Direct Context-Aware Modeling for End-to-End Monaural Speech Separation,” in INTERSPEECH, 2020.
  * [7] C. Subakan, M. Ravanelli, S. Cornell, M. Bronzi, and J. Zhong, “Attention Is All You Need In Speech Separation,” in ICASSP, 2021.
  * [8] M. Maciejewski, G. Wichern, E. McQuinn, and J. L. Roux, “WHAMR!: Noisy and reverberant single-channel speech separation,” in ICASSP, 2020.
  * [9] T. Cord-Landwehr, C. Boeddeker, T. v. Neumann, C. Zorila, R. Doddipatla, and R. Haeb-Umbach, ”Monaural source separation: From anechoic to reverberant enviorments,” arXiv preprint arXiv:2111.07578v2, 2022.
  * [10] C. Subakan, M. Ravanelli, S. Cornell, F. Grondin, and M. Bronzi, “On using transformers for speech-separation,” arXiv preprint arXiv:2202.02884, 2022.
  * [11] K. Ho, J. Hung, and B. Chen, ”Bi-Sep: A Multi-Resolution Cross-Domain Monaural Speech Separation Framework,” in TAAI, 2022.
  * [12] F. A. Chao, J. W. Hung, and B. Chen, ”Cross-Domain Single-Channel Speech Enhancement Model with BI-Projection Fusion Module for Noise-Robust ASR,” in ICME, 2021.
  * [13] W. Ravenscroft, S. Goetze, and T. Hain, ”Utterance Weighted Multi-Dilation Temporal Convolutional Networks for Monaural Speech Dereverberation,” in IWAENC, 2022.
  * [14] J. L. Roux, S. Wisdom, H. Erdogan, and J. R. Hershey, “SDR – Half-baked or Well Done?” in ICASSP, 2019.
  * [15] S. Wisdom, E. Tzinis, H. Erdogan, R. Weiss, K. Wilson, and J. Hershey, “Unsupervised sound separation using mixture invariant training,” in Advances in Neural Information Processing Systems, 2020.
  * [16] J. Heitkaemper, D. Jakobeit, C. Boeddeker, L. Drude, and R. Haeb-Umbach, “Demystifying TasNet: A dissecting approach,” in ICASSP, 2020.
  * [17] D. Ditter and T. Gerkmann, “A Multi-Phase Gammatone Filterbank for Speech Separation Via Tasnet,” in ICASSP, 2020.
  * [18] Z. Q. Wang, G. Wichern, and J. L. Roux, ”On the Compensation Between Magnitude and Phase in Speech Separation,” IEEE Signal Process. Lett., vol. 28, pp. 2018-2022, 2021.
  * [19] Y. Koyama, T. Vuong, S. Uhlich, and B. Raj ”Exploring the Best Loss Function for DNN-Based Low-latency Speech Enhancement with Temporal Convolutional Networks,” arXiv preprint arXiv:2005.11611v3, 2020.
  * [20] D. de Oliveira, T. Peer, and T.Gerkmann ”Efficient Transformer-based Speech Enhancement Using Long Frames and STFT Magnitudes,” in INTERSPEECH, 2022.
  * [21] Y. Avargel and I. Cohen, “On multiplicative transfer function approximation in the short-time fourier transform domain,” IEEE Signal Process. Lett., vol. 14, no. 5, pp. 337–340, 2007.
  * [22] D. Wang and J. Lim, “The unimportance of phase in speech enhancement,” IEEE Trans. Audio Speech Lang. Process., vol. 30, no. 4, pp. 679–681, Aug. 1982.
  * [23] T. Peer and T. Gerkmann, “Intelligibility Prediction of Speech Reconstructed From Its Magnitude or Phase,” in ITG Conference on Speech Communication, 2021.
  * [24] T. Peer and T. Gerkmann, “Phase-aware deep speech enhancement: It’s all about the frame length,” arXiv preprint arXiv:2203.16222, 2022.
  * [25] J. Chen, Z. Wang, D. Tuo, Z. Wu, S. Kang, and H. Meng, ”FullSubNet+: Channel Attention Fullsubnet with Complex Spectrograms for Speech Enhancement,” in ICASSP, 2022.
  * [26] Y. Tsao, K. Ho, J. Hung, and B. Chen, ”Adaptive-FSN: Integrating full-band extraction and adaptive sub-band encoding for monaural speech enhancement,” in SLT, 2022.
  * [27] N. Takahashi and Y.Mitsufuji, “Multi-scale multi-band densenets for audio source separation,” in WASPAA, 2017.
  * [28] Rong Chao, Cheng Yu, Szu-Wei Fu, Xugang Lu, and Yu Tsao, “Perceptual contrast stretching on target feature for speech enhancement,” in INTERSPEECH, 2022.
  * [29] E. Perez, F. Strub, H. De Vries, V. Dumoulin, and A. Courville, “Film: Visual reasoning with a general conditioning layer,” in AAAI-18, 2018.
  * [30] R. Xiong et al., ”On layer normalization in the transformer architecture,” in ICML, 2020.
  * [31] G. Wichern et al., ”Wham!: Extending speech separation to noisy environments,” in INTERSPEECH, 2019.


# Scattering of $W$ and $Z$ Bosons
at High-Energy Lepton Colliders

Christian Flepera, Wolfgang Kiliana, Jürgen Reuterb, Marco Sekullac
christian.fleper<EMAIL_ADDRESS>siegen.dejuergen.reuter<EMAIL_ADDRESS>
###### ABSTRACT

We present a new study of quasi-elastic $W$ and $Z$ scattering processes in
high-energy $e^{+}e^{-}$ collisions, based on and extrapolating the low-energy
effective theory which extends the Standard Model with a $125\;\text{GeV}$
Higgs boson. Besides parameterizing deviations in terms of the dimension-8
operators that arise in the effective theory, we also study simplified models
of new physics in $W/Z$ scattering in terms of scalar and tensor resonance
multiplets. The high-energy asymptotics of all models is regulated by a
universal unitarization procedure. This enables us to provide benchmark
scenarios which can be meaningfully evaluated off-shell and in exclusive event
samples, and to determine the sensitivity of an $e^{+}e^{-}$ collider to the
model parameters. We analyze the longitudinal vector boson scattering modes,
where we optimize the cuts for the fiducial cross section for different
collider scenarios. Here, we choose energy stages of 1.0, 1.4 and 3 TeV, as
motivated by the extendability of the ILC project and the staging scenario of
the CLIC project.

###### Contents

  1. 1 Introduction
  2. 2 Effective Field Theory and Vector Boson Scattering
  3. 3 Anomalous Interactions of Vector Bosons
  4. 4 Unitary Extrapolation
  5. 5 Cut-Based Extraction of the VBS Signal
  6. 6 Results for EFT Operator Coefficients
  7. 7 Results for Resonances
  8. 8 Conclusions
  9. A Dimension-8 Operators that Affect VBS

##  1 Introduction

Quasi-elastic scattering processes of the massive electroweak bosons
$W^{\pm},Z$ (vector-boson scattering, VBS) are a cornerstone in the
phenomenology of electroweak interactions. The longitudinal polarization
components of on-shell energetic $W$ and $Z$ particles are closely related to
the unobservable Goldstone bosons that constitute the elementary Higgs doublet
together with the physical Higgs boson [1, 2]. Their interactions thus probe
the mechanism of electroweak symmetry breaking.

If the Higgs was missing from the particle spectrum, the $W/Z$ interaction
strength would rise with energy into a non-perturbative regime, indicating an
intrinsic cutoff of the effective theory and new strong interactions [3, 4].
However, the recent discovery of a $125\;\text{GeV}$ Higgs boson implies that
this is not the case. The existence of the Higgs boson allows all VBS
interactions to remain weak asymptotically, calculable in electroweak
perturbation theory. For the necessary cancellation of terms to take effect to
all orders, the vector-boson, Higgs and Goldstone couplings must coincide
exactly with their Standard Model (SM) values [5].

Without sufficient experimental data, we cannot decide whether the pure SM is
the correct description of the electroweak symmetry-breaking sector. Beyond
the SM, processes that involve Higgs and Goldstone fields could rather open a
new portal to phenomena and structures that do not couple directly to matter
particles, and are thus detached from immediate experimental access. New
physics in the Higgs-Goldstone sector could have only a tiny impact on low-
energy precision observables and is therefore largely unconstrained by
existing data. Collider experiments will have to search for new effects in
this area and complete our knowledge about the particle and interaction
spectrum at accessible energies.

VBS processes have been observed at the Large Hadron Collider (LHC) in Run I
[6, 7, 8], and the analysis of future LHC runs at full energy and increased
luminosity will considerably improve our knowledge in this sector. At hadron
colliders, VBS is accessible in processes of the type $pp\to
q^{\ast}q^{\ast}\to qq+VV$, where $q$ indicates any light quark, and $V=W,Z$.
The analysis has been demonstrated to be feasible, but nevertheless suffers
from limitations that do not simply disappear with increasing collider
luminosity or energy. Leptonic decays of vector bosons either yield incomplete
kinematic information ($W$) or a low branching ratio ($Z$). Hadronic decays
are difficult to isolate from a large QCD background. VBS processes must be
separated from vector-boson pair production, $q\bar{q}^{\prime}\to VV$, QCD
production of $VV$ and two jets, as well as top-quark production as SM
background. The initial state, i.e., the flavor and energy of the initial
quarks, cannot be controlled or detected, and steeply falling parton structure
functions limit the accessible energy for the elementary VBS interaction.

By contrast, $e^{+}e^{-}$ colliders provide a clean environment where the VBS
process class has a unique signature, and most decays of $W$ and $Z$ bosons
are accessible and can be observed with high efficiency and purity [9, 10].
The initial state is known exactly, and the $e^{+}e^{-}$ c.m. energy is fixed
up to minor electromagnetic radiation effects. We expect more detailed and
complementary information to be available from $e^{+}e^{-}$ collisions
compared to hadronic collisions. The decisive factor for an $e^{+}e^{-}$
collider is the availability of sufficient energy combined with high
luminosity.

The complete list of $e^{+}e^{-}$ VBS processes accessible at an $e^{+}e^{-}$
collider with sufficient energy reads

$\displaystyle e^{+}e^{-}$ $\displaystyle\to\bar{\nu}_{e}\nu_{e}W^{+}W^{-},$
(1) $\displaystyle\quad\bar{\nu}_{e}\nu_{e}ZZ,$ (2)
$\displaystyle\quad\bar{\nu}_{e}e^{-}W^{+}Z,$ (3) $\displaystyle\quad
e^{+}\nu_{e}ZW^{-},$ (4) $\displaystyle\quad e^{+}e^{-}W^{+}W^{-},$ (5)
$\displaystyle\quad e^{+}e^{-}ZZ.$ (6)

with various decay channels of the final-state bosons. These processes allow
us to study interactions of $W$ and $Z$ bosons in the following elementary
scattering channels, which are realized as distinct, approximately on-shell
factorized contributions to the amplitudes:

$\displaystyle W^{+}W^{-}$ $\displaystyle\to W^{+}W^{-},$ (7) $\displaystyle
W^{+}W^{-}$ $\displaystyle\to ZZ,$ (8) $\displaystyle W^{\pm}Z$
$\displaystyle\to W^{\pm}Z,$ (9) $\displaystyle ZZ$ $\displaystyle\to
W^{+}W^{-},$ (10) $\displaystyle ZZ$ $\displaystyle\to ZZ.$ (11)

By kinematics alone, the threshold for these processes is just above the
vector-boson pair production threshold of $160\ldots 180\,\text{GeV}$, but for
a meaningful analysis of deviations from the SM, a significantly larger
vector-boson pair energy (i.e., invariant mass of the $VV$ system) is
necessary.

The prospects for the measurement of VBS processes at lepton colliders,
including potential non-SM contributions, have been studied extensively in the
literature [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. However, most previous
studies have investigated no-Higgs (or heavy-Higgs) scenarios, occasionally in
comparison to the pure SM with a light Higgs of some arbitrarily assumed mass.
Since a Higgs-like boson has been found, its mass has been precisely
determined, and the measurement of its couplings is in accordance with the
pure SM prediction, Higgs-less models do no longer provide a viable scenario
for electroweak interactions. The analysis of VBS should rather be based on
models that reduce to the SM with a $125\;\text{GeV}$ Higgs at low energy and
take into account all recently accumulated knowledge about the Higgs
boson.111Turning this around, VBS at high energy can be utilized to supply
indirect information on Higgs boson properties [21].

In this paper, we therefore present a new study of VBS processes at an
$e^{+}e^{-}$ lepton collider. We take the SM with a Higgs mass of
$125\;\text{GeV}$ as reference and determine the sensitivity to effects beyond
the SM in VBS interactions. Given the exploratory nature of this task and the
emergence of strong-interaction effects, we can restrict our calculation to
leading order in the SM and EFT predictions. Actual data analysis should
incorporate NLO corrections in the SM and its EFT extension, cf. [22, 23, 24],
which eventually should become available for the complete six-fermion partonic
processes. For a meaningful extrapolation to energy and parameter ranges where
partial-wave unitarity becomes an issue, we apply a universal unitarization
method, the T-matrix framework. The concrete results are computed for specific
values of the collider energy, $\sqrt{s}=1.4\;\text{TeV}$ and
$3.0\;\text{TeV}$ with integrated luminosities of
$\mathcal{L}_{int}=~{}1.5\;\text{ab}^{-1}$ and $2\;\text{ab}^{-1}$, as planned
for the Compact Linear Collider (CLIC) [25, 26, 27]. Furthermore, we make use
of specific properties of the CLIC collider environment in its currently
planned state, including the most relevant detector properties, to estimate
the sensitivity on anomalous effects beyond the SM. For completeness, we also
include numbers for a lower collider energy of $1\;\text{TeV}$ with an
integrated luminosity of $\mathcal{L}_{int}=~{}5\;\text{ab}^{-1}$, which
should also illuminate the potential of an energy upgrade of the International
Linear Collider (ILC) [28, 29].

##  2 Effective Field Theory and Vector Boson Scattering

The theoretical basis for the current study is the SM with a single complex
Higgs doublet. Since we want to provide not just the unique SM prediction but
a range of possibilities for the high-energy behavior of electroweak
interactions, we have to regard the SM as an effective field theory (EFT).
I.e., we assume an infinite series of interactions, organized by operator
dimension [30, 31, 32]. The pure SM limit consists of setting the couplings of
all operators with dimension greater than four to zero, or alternatively,
letting the intrinsic mass scale of those dimensionful couplings to infinity.

Within the EFT formalism, we assume that the strong-electroweak
$SU(3)_{QCD}\times SU(2)_{L}\times U(1)_{Y}$ gauge symmetry of the SM
Lagrangian is a fundamental property, and therefore organize all higher-
dimensional operators in terms of gauge-invariant polynomials. This implies
global strong-electroweak invariance for all terms and promoting partial
derivatives to covariant derivatives where required by the field
representation. We do not consider CP symmetry breaking effects (beyond those
already present in the SM) in the current study.

The SM Higgs, in the gauge-invariant Lagrangian, appears as a complex doublet
which transforms linearly under $SU(2)_{L}\times U(1)_{Y}$.222A non-linear
Higgs representation might be chosen to allow for more freedom in Higgs
couplings that are not yet constrained with precision, but this property is of
secondary importance in our study of VBS processes. It becomes more relevant
when considering also Higgs final states, cf. [33]. If we neglect the
interactions of heavy SM fermions, the global symmetry is approximately
$SU(2)_{L}\times SU(2)_{R}$ [34, 35]. This symmetry is explicitly broken by
the hypercharge gauge coupling. The obstruction can be controlled as a
spurion, and the analysis can be based on an exactly $SU(2)_{L}\times
SU(2)_{R}$ symmetric Lagrangian. This is a reasonable simplification since we
are interested in the high-energy range of VBS processes where gauge
interactions play a minor role, i.e., can be viewed as a perturbation
entirely. The Higgs vacuum expectation value (vev) spontaneously breaks the
global symmetry down to the diagonal $SU(2)_{L+R}$, known as the custodial
symmetry $SU(2)_{C}$.

In this context, it is natural to represent the Higgs field multiplet as a
$2\times 2$ matrix [36, 37],

$\displaystyle\mathbf{H}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}v+h-\mathrm{i}w^{3}&-\mathrm{i}\sqrt{2}w^{+}\\\
-\mathrm{i}\sqrt{2}w^{-}&v+h+\mathrm{i}w^{3}\\\ \end{pmatrix}\quad.$ (12)

which transforms linearly under $U\in SU(2)_{L}$ and $V\in SU(2)_{R}$ as

$\mathbf{H}\to U\mathbf{H}V^{\dagger}\quad.$ (13)

The covariant derivative of the Higgs matrix is given by

$\mathbf{D}_{\mu}\mathbf{H}=\partial_{\mu}\mathbf{H}-ig\mathbf{W}_{\mu}\mathbf{H}+ig^{\prime}\mathbf{H}\mathbf{B}_{\mu}$
(14)

where

$\mathbf{W}_{\mu}\equiv
W_{\mu}^{a}\frac{\tau^{a}}{2},\qquad\mathbf{B}_{\mu}\equiv
B_{\mu}\frac{\mathbf{T}}{2}$ (15)

and $\mathbf{T}$ is a $SU(2)_{R}$-breaking spurion with ground state
$\mathbf{T}=\tau^{3}$. Finally, we define the matrix-valued field strengths

$\mathbf{W}^{\mu\nu}=\partial_{\mu}\mathbf{W}_{\nu}-\partial_{\nu}\mathbf{W}_{\mu}-\mathrm{i}g\left[\mathbf{W}_{\mu},\mathbf{W}_{\nu}\right],\quad\mathbf{B}^{\mu\nu}=\partial_{\mu}\mathbf{B}_{\nu}-\partial_{\nu}\mathbf{B}_{\mu}$
(16)

and quote the bosonic part of the SM Lagrangian (omitting QCD),

$\displaystyle\begin{aligned}
\mathcal{L}_{SM}=&-\frac{1}{2}\operatorname{tr}\left[\mathbf{W}_{\mu\nu}\mathbf{W}^{\mu\nu}\right]-\frac{1}{2}\operatorname{tr}\left[\mathbf{B}_{\mu\nu}\mathbf{B}^{\mu\nu}\right]\\\
&+\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\mu}\mathbf{H}\right]+\mu^{2}\operatorname{tr}\left[\mathbf{H}^{\dagger}\mathbf{H}\right]-\frac{\lambda}{2}\left(\operatorname{tr}\left[\mathbf{H}^{\dagger}\mathbf{H}\right]\right)^{2}\end{aligned}\quad.$
(17)

The power of an EFT series as a perturbative expansion lies in the accuracy of
a truncation at low order. New mutual and self-interactions of bosons only
take the form of gauge-invariant operators of even dimension. The leading non-
SM order is thus dimension six, so it appears natural to truncate the EFT at
this order. The complete set of dimension-six operators has been discussed
extensively in the literature [38, 39, 40, 41, 42, 43].

However, new-physics effects in the Higgs-Goldstone sector, including new
strong interactions, largely decouple from precision observables as they are
accessible today. While several dimension-six operators in the EFT do affect
VBS processes at tree level, they simultaneously modify other couplings which
can be measured independently. Such operators are an inadequate representation
of the specific new-physics scenarios that VBS is most sensitive to. In fact,
for the purpose of studying VBS, we may assume that the coefficients of
dimension-six operators are known to sufficient precision; we set them to zero
as the standard reference point.

Traces of genuine new Higgs-Goldstone physics appear first in dimension-eight
operators. These modify VBS interactions independently of other types of
interaction. For instance, the exchange of massive resonances generically
results in dimension-eight effective operators. Our study therefore includes
the relevant dimension-eight operators.

It is evident that any truncation at this level, while technically consistent,
is of questionable value. Amplitudes modified by dimension-eight operators
rise rapidly with energy and thus yield large effects [37]. This fact also
indicates the breakdown of the perturbative expansion. Our version of the EFT
with dimension-eight operators is therefore not intended as an approximation
that can be systematically improved. It rather serves as a phenomenological
tool to describe dominant non-SM phenomena in VBS. At low energy,
$O(100\;\text{GeV})$, it smoothly matches to the EFT within its range of
validity. Extrapolated to high energy, it gives a flavor of the maximum impact
that BSM can have on VBS. The extrapolation incorporates unitarity as the only
damping mechanism and thus asymptotically approaches the unitarity bound for
each partial-wave amplitude. However, additional structure can easily be
included, as we will describe below using simplified models containing
resonances.

##  3 Anomalous Interactions of Vector Bosons

The complete set of dimension-eight operators for the SM fields is rather
large. In line with the above considerations, we resort to standard
simplifications in order to make the set manageable. First of all, as stated
before, we only consider bosonic operators. (That is, we treat fermionic
currents as external probes for bosonic interactions, neglecting genuine
fermionic anomalous contributions.) We furthermore assume the global
$SU(2)_{L}\times SU(2)_{R}$ symmetry of the fermion- and gaugeless SM to hold
also at higher orders. This introduces relations between $W$ and $Z$
scattering amplitudes and simplifies the scattering matrix for the purpose of
unitary extrapolation.

Focusing on operators that directly modify quartic vector-boson interactions,
there are three classes which affect Higgs/Goldstone bosons only (index: $S$),
gauge bosons only (index: $T$), or both (index: $M$), respectively. We recall
that, taking EWSB into account, Goldstone bosons are probed via the
longitudinal polarization direction of energetic $W$ and $Z$ bosons, while
gauge bosons translate into transversal polarization.

Operators involving only the Higgs/Goldstone bosons sector, i.e., S-type, are
represented by a combination of covariant derivatives acting on the Higgs
fields. Two linearly independent operators can be identified

$\displaystyle\mathcal{L}_{S,0}$ $\displaystyle=$ $\displaystyle F_{S,0}\ $
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\mathbf{D}_{\nu}\mathbf{H}\right]\operatorname{tr}\left[\left(\mathbf{D}^{\mu}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\nu}\mathbf{H}\right],$
(18a) $\displaystyle\mathcal{L}_{S,1}$ $\displaystyle=$ $\displaystyle
F_{S,1}\ $
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\mu}\mathbf{H}\right]\operatorname{tr}\left[\left(\mathbf{D}_{\nu}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\nu}\mathbf{H}\right].$
(18b)

In the present paper, we do not study the transversal and mixed interactions
but defer this to future work. For completeness, we list the $T$ and $M$-type
operators in the Appendix. We also neglect operators which are proportional to
$\mathbf{H}^{\dagger}\mathbf{H}$; as long as Higgs final states are not
considered, this class of operators merely renormalizes the dimension-six part
of the EFT Lagrangian.

We may compare this EFT framework with the corresponding expansion [44, 45]
and subsequent analysis for the no-Higgs (or heavy-Higgs) case, as it was
investigated, for instance, in [16, 17]. In the pure SM, a light Higgs boson
and no higher-order operators, the amplitude for VBS remains perturbative at
all energies. Beyond the SM, new physics that appears exclusively in the
Higgs-Goldstone sector corresponds to dimension-eight operators as argued
above, therefore any deviation from the SM increases rapidly with energy.
Since the pure SM amplitude is small, interference plays a minor role and the
phenomenologically relevant behavior originates from the dimension-eight
interactions, squared. By contrast, if the light Higgs boson did not exist,
the reference amplitude would grow with energy and render the interaction
strong in the TeV range by itself, while anomalous effects would contribute
first via their interference, which is a less striking modification of the
reference amplitude. In this study, which is based on the now established
light-Higgs scenario, we therefore will face a sudden transition from a
weakly-interacting to a strongly-interacting regime when increasing the energy
in the VBS process.

##  4 Unitary Extrapolation

The SM amplitudes for VBS, as for any other elementary process, are
perturbative throughout the accessible energy range and therefore do not pose
a unitarity problem. If anomalous effects are present which cannot simply be
mapped to an extended renormalizable (weakly interacting) model, this property
is lost, and we have to deal with unitarity violation in the tree-level
prediction. Clearly, this indicates the breakdown of perturbation theory. On
the other hand, the quantitative analysis of actual data relies on the
availability of a prediction which depends on a suitable set of new
parameters. Such a reference allows us to quantitatively study or exclude a
deviation from the SM. In order to discard grossly unphysical
parameterizations, we have to ensure that any such extrapolation is at least
in accordance with unitarity, all assumed symmetries, and smoothly matches to
the low-energy effective theory

In [37, 46], based on earlier work in [47], we have described the T-matrix
unitarization framework as a generic scheme that allows for a unitary
extrapolation of any model without introducing arbitrary artefacts in the
asymptotic regime. We adopt this framework for the current paper.

At the fundamental level of the complete scattering matrix
$\mathbf{S}=1+\mathrm{i}\mathbf{T}$ (T matrix), the unitarization framework
relies on the formula

$\displaystyle\mathbf{T}(\mathbf{T}_{0})$
$\displaystyle=\frac{1}{\mathrm{Re}\left({\mathbf{T}_{0}}^{-1}\right)-\frac{\mathrm{i}}{2}\mathds{1}}\,,$
(19)

which transforms an arbitrary model of the scattering matrix $\mathbf{T}_{0}$
into a unitary model of the scattering matrix $\mathbf{T}$. In particular, the
set of amplitudes is invariant under the transformation if it already respects
unitarity. For the application of the formula, it is advantageous to
diagonalize the scattering matrix first and apply unitarization to
eigenamplitudes, since this amounts to a simple multiplication or subtraction.
In the present context, this can be done most easily in the high-energy limit
where the gauge sector decouples from the Higgs sector. Since unitarity is an
issue only for high energies, such an approach is sufficient to remove all
dangerous terms.

The method does not refer to any property of the original model which is
encoded in $\mathbf{T}_{0}$; in particular, it does not assume a perturbative
expansion or a particular analytical structure. While it reconciles any model
$\mathbf{T}_{0}$ with unitarity, the result is not a unique prediction. It is
merely a model that describes possible behavior of the amplitude up to the
highest energies, as opposed to a unitarity-violating model or extrapolation
that describes impossible behavior.

In practice, we identify the asymptotically leading terms in the diagonal
scattering matrix, replace them by their unitary equivalents, and invert the
diagonalization. Factoring out Lorentz tensors and subtracting the original
EFT contribution, we cast this in the form of extra momentum-dependent Feynman
rules. The new terms resemble the local Feynman rules that describe the
original EFT operators, but they carry prefectors that depend on invariant
momentum combinations in a non-analytical form. Nevertheless, for
calculational purposes the new terms play a role analogous to the local
counterterms that arise in a NLO calculation, and are straightforward to take
into account in the construction and evaluation of scattering matrix elements.
This allows us to perform off-shell calculations and evaluate the unitarized
amplitudes for external fermions, as long as the kinematic assumptions
underlying the procedure are satisfied.

For illustration, in plots below (Figs. 4, 5, 6) we display both the
unitarized model behavior and the unphysical results that we would have
obtained without unitarization. The latter are marked as dashed lines, while
continous curves refer to the respective unitary versions. While for some
parameter sets, the unitarization correction remains a minor issue, there are
various cases where unitarization has a large impact. This property clearly
indicates that for CLIC energies, VBS processes are probed in a range where
undetermined new-physics effects are actually important, and systematic
approximations do no longer yield unambiguous results. Under no circumstances,
calculation results without unitarization may be used for the analysis of
actual data, as this would grossly overestimate the sensitivity on the model
parameters.

##  5 Cut-Based Extraction of the VBS Signal

Based on the theoretical framework as described above, we now turn to the
actual prospects for measuring vector-boson scattering processes at high
effective energy at a high-energy lepton collider, like e.g. the CLIC
collider. The strategy of such an analysis does not depend strongly on the
underlying physics model. We can therefore follow the ideas of [13, 16, 17]
and adapt the analysis to the CLIC environment.

Figure 1: Signal Feynman diagrams contributing to the VBS process.

The main idea is to isolate a signal of vector-boson pairs in association with
very forward neutrinos or electrons. This is the kinematic situation where in
the diagrams of Fig. 1, the incoming vector bosons which participate in the
VBS interaction have small invariant mass, so the amplitude is enhanced by the
small denominator of the t-channel propagators. The nonvanishing mass of
$V=W,Z$ bosons cuts off the approach to the actual particle pole, so the
enhancement factor is of order $m_{V}^{2}/E_{V}^{2}$ in this kinematic range,
and the typical recoil $p_{T}$ of the radiating lepton becomes of order
$m_{V}$. Graphs without the VBS interaction (Figs. 2, 3) do not benefit from
this effect.

Figure 2: Feynman diagrams contributing to the irreducible background. Figure
3: Feynman diagrams contributing to the partially reducible background.

A large part of the total rate is due to $W$-boson pair production initiated
by photons, cf. Fig. 3 (left). Photon-induced VBS processes are interesting by
themselves, but photons do not have a longitudinal component and therefore do
not receive contributions from new physics in the Higgs sector. For our
purposes, the photon-induced contribution is therefore considered as a
background. In contrast to the distribution of intermediate $W/Z$ bosons,
which is dominated by $Q\sim m_{W/Z}$, the distribution of photons extends
down to nearly zero momentum transfer and is cut off only by the finite
electron mass. Electrons in forward direction disappear in the beampipe and
are thus indistinguishable from neutrinos, so in this kinematic range,
intermediate photons cannot be separated by detecting the radiating fermion.
Nevertheless, this background can be reduced without losing too much of the
signal by vetoing against very forward electrons and simultaneously cutting on
the $p_{T}$ distribution of the vector-boson pair system.

We are interested in the vector-boson pair system at large combined invariant
mass, as this is the energy value that enters the basic VBS interaction.
Hence, we propose the following set of selection cuts, where the first number
always refers to the 3.0 TeV CLIC staging, while the one in parentheses
corresponds to the lower-energy staging at 1.4 TeV.

  1. 1.

$M_{inv}(\bar{\nu}\nu)>230(175)\;\text{GeV}$. The signal process contains two
neutrinos in the final state. This cut removes events where the neutrinos
originate from Z decay; a majority of these neutrinos will have an invariant
mass $M_{inv}(\bar{\nu}\nu)\sim 91\;\text{GeV}$. Backgrounds from $W^{+}W^{-}$
and QCD four-jet production are also removed by this cut.

  2. 2.

$|\cos\theta(W/Z)|<0.8$ and $p_{\perp}(W/Z)>300(180)\;\text{GeV}$. Events
where the vector bosons have a small distance from the beam pipe or a small
transverse momentum are cut away. This reduces backgrounds which result from
t-channel exchange in the subprocess.

  3. 3.

$\theta(e)>15$ mrad and $p_{\perp}(WW)>100(50)\;\text{GeV}$,
$p_{\perp}(ZZ)>60(40)\;\text{GeV}$. Together with a cut on the transverse
momentum of the vector boson pair system, background resulting from
$\gamma\gamma$ fusion will substantially be decreased.

  4. 4.

$900(800)\;\text{GeV}<M_{inv}(WW)<1900(1175)\;\text{GeV}$,
$850(800)\;\text{GeV}<M_{inv}(ZZ)<1900(1175)$ GeV. The influence of the
operators Eq. (18) increases with the invariant mass of the vector boson pair.
We therefore only consider events within a small invariant mass range.

The following cuts are applied for the ILC staging scenario at
$1.0\,\text{TeV}$:

  1. 1.

$M_{inv}(\bar{\nu}\nu)>150\;\text{GeV}$.

  2. 2.

$|\cos\theta(W/Z)|<0.8$ and $p_{\perp}(W/Z)>150\;\text{GeV}$.

  3. 3.

$\theta(e)>15$ mrad and $p_{\perp}(WW)>45\;\text{GeV}$,
$p_{\perp}(ZZ)>40\;\text{GeV}$.

  4. 4.

$575\;\text{GeV}<M_{inv}(WW)<800\;\text{GeV}$,
$600\;\text{GeV}<M_{inv}(ZZ)<800$ GeV.

For the calculation, we have used the implementation of the SM in the WHIZARD
Monte-Carlo generator [48, 49]. The SM extensions that we consider in later
sections are likewise handled by dedicated WHIZARD model implementations,
taking into account unitarization corrections where necessary. The WHIZARD
generator provides a physics simulation framework that can include beam
properties (polarization, beamstrahlung and ISR), the complete partonic final
state to leading and next-to-leading order QCD, parton shower, and
hadronization, eventually complemented by detector-level tools and analysis
methods [50, 51, 52, 53].

For this exploratory study, we have largely restricted ourselves to on-shell
$W/Z$ bosons in the final state and concentrate on total cross section values
after cuts. Polarization, ISR, and beamstrahlung are not taken into account
for the tables below. Further below, we comment on possible improvements if
polarization and details of the partonic final state are taken into account,
using off-shell simulation for the latter. We expect that a future
experimental study which can be performed using the WHIZARD generator in
connection with detector simulation, jet algorithms, etc., should give more
accurate results which we nevertheless expect to lie in the numerical range as
our final results suggest. Clearly, especially ISR and beamstrahlung will lead
to a distortion of distributions and to a depletion of events in the high-
energy region, but we do not expect our results to fundamentally change. Our
first estimates which should be backed up by future experimental studies
confirmed that, and especially the number that roughly half of the events is
in the highest ten-percent energy bin from Tab. 2.2 in [25] when beamstrahlung
is actually simulated.

The tables 1 and 2 contain our results for the SM cross sections of the
processes of interest, before and after cuts, respectively. As argued above,
the results apply to on-shell bosons in the final state and do not refer to
detailed properties of the CLIC collider, with two notable exceptions that we
discuss in the following.

Process | 1.4 TeV | 3 TeV | Factor
---|---|---|---
$W^{+}W^{-}\nu\bar{\nu}$ | 47.1 | 132 | 1
$W^{+}W^{-}e^{+}e^{-}$ | 1570 | 3820 | 1
$W^{\pm}Ze^{\mp}\nu$ | 138 | 408 | 0.136
$ZZe^{+}e^{-}$ | 3.78 | 4.70 | 0.019
$W^{+}W^{-}(Z\to\nu\bar{\nu})$ | 11.7 | 9.35 | 1
$ZZ\nu\bar{\nu}$ | 15.7 | 57.5 | 1
$ZZe^{+}e^{-}$ | 3.78 | 4.70 | 1
$W^{\pm}Ze^{\mp}\nu$ | 138 | 408 | 0.136
$W^{+}W^{-}e^{+}e^{-}$ | 1570 | 3820 | 0.019
$ZZ(Z\to\nu\bar{\nu})$ | 0.484 | 0.237 | 1
Table 1: Standard model total cross sections in fb ($\pm 1\%$ error) without cuts for center-of-mass energies of $\sqrt{s}=1.4\;\text{TeV}$ and $3\;\text{TeV}$. Both particle beams are unpolarized. Detection efficiencies and branching ratios are not included. All cross sections have to be multiplied by the factors in the fourth column to take the misidentification of vector bosons into account. Process | 1.4 TeV | 3 TeV | Factor
---|---|---|---
$W^{+}W^{-}\nu\bar{\nu}$ | 0.119 | 0.790 | 1
$W^{+}W^{-}e^{+}e^{-}$ | 0.000 | 0.000 | 1
$W^{\pm}Ze^{\mp}\nu$ | 0.269 | 1.200 | 0.136
$ZZe^{+}e^{-}$ | 0.000 | 0.000 | 0.019
$W^{+}W^{-}(Z\to\nu\bar{\nu})$ | 0.039 | 0.610 | 1
$ZZ\nu\bar{\nu}$ | 0.084 | 0.790 | 1
$ZZe^{+}e^{-}$ | 0.000 | 0.000 | 1
$W^{\pm}Ze^{\mp}\nu$ | 0.288 | 1.590 | 0.136
$W^{+}W^{-}e^{+}e^{-}$ | 0.000 | 0.000 | 0.019
$ZZ(Z\to\nu\bar{\nu})$ | 0.000 | 0.000 | 1
Table 2: Same as Tab. 1, but with cuts.

  1. 1.

For suppressing the contribution from photon-induced processes, the analysis
has to rely on the ability to detect energetic electrons in the very forward
region, which are barely deflected by the photon emission. If such electrons
are vetoed against, the ambiguity between $W$ and $Z/\gamma$ bosons in the
initial state is greatly reduced. For the current study we have assumed that a
veto against electrons is possible down to an angle of 15 mrad [54].

  2. 2.

The clean environment and triggerless operation of a lepton collider allows us
to detect final-state vector bosons by their decay products in essentially all
channels. For the current study, we have concentrated on the hadronic decay
channels. These decays provide the major part of the decay branching fractions
and yield complete kinematic information. In particular, the momenta of the
forward neutrinos can be inferred from the missing energy and momentum. A
disadvantage of hadronic channels is the absence of charge information and the
finite jet-pair invariant mass resolution, which adds on the natural decay
width of $W$ and $Z$ bosons. As a result, there is a probability for
misidentification between $W$ and $Z$. We take this into account by the matrix
for identification probabilities as it was determined in [55]:

$\displaystyle W$ $\displaystyle\to 88\%\ W,\ 12\%\ Z$ (20a) $\displaystyle Z$
$\displaystyle\to 12\%\ W,\ 88\%\ Z$ (20b)

Partonic $WW$, $WZ$ and $ZZ$ final states therefore will be identified as a
$WW(ZZ)$ event with probabilities 77.4%, 10.6%, 1.4%, which yields the
weighting factors 1:0.136:0.019 given in the final column of both tables 1 and
2. Taking the hadronic $W$ and $Z$ boson branching ratios of 67.70% and 69.91%
into account and including the di-lepton modes of the Z boson (BR=6.729% for
$e^{+}e^{-}$ and $\mu^{+}\mu^{-}$), the efficiencies for detecting a $WW$,
$WZ$, $ZZ$ pair originating from a partonic $WW$, $WZ$, $ZZ$ final state are
35.4%, 40.1% and 45.5%, respectively.

A significant part of the $VV\nu\nu$ final state is contributed by triple
vector boson production, where the third boson is a $Z$ with invisible decay
to neutrinos, cf. Fig. 3 (right). This might be considered as a separate
background, and it is tempting to calculate this via a separate calculation of
triple vector boson production with $Z$ decay. However, the process is probed
in a kinematical region where the would-be $Z$ boson is far off shell. The
distribution in this region cannot be defined in a gauge-invariant way if we
select only diagrams with a virtual $Z$ boson. We have compared the results of
[16, 17], which were obtained for this selection of Feynman diagrams using
’tHooft-Feynman gauge for the gauge bosons, with an analogous WHIZARD
calculation where all amplitudes are computed in unitarity gauge. As one may
expect, the two results differ by a large factor in the signal region of high
$\bar{\nu}\nu$ invariant masses, while coinciding on the $Z$ mass peak. The
momentum factors in the unitarity-gauge propagators produce a gauge-dependent
excess which is cancelled against a matching piece if the full gauge-invariant
set of diagrams is taken into account. In tables 1 and 2, we quote the
unitarity-gauge results for the three-boson subprocess for completeness, but
we emphasize that after cuts, this number is unphysical and actually
irrelevant for the signal sensitivity; as a background, it can be ignored for
all practical purposes.

In other words, such a selection of Feynman graphs should not be attempted and
a triple-boson background contribution cannot be defined. For the remainder of
our study we only use results for the complete process which are gauge
invariant by construction.

Besides gauge-invariance issues, we should note that the $VVZ$ contribution to
the processes of interest does not depend significantly on the coefficients of
the operators in Eq. 18. This is easily understood if we remember that in the
high-energy limit gauge and Goldstone bosons decouple. An intermediate off-
shell $Z$ or $\gamma$ couples to a massless fermion current and therefore
behaves as a gauge boson. A Goldstone boson contribution is excited only via
mixing, suppressed by $m_{Z}/\sqrt{s}$. Indeed we observe that in the low-mass
region for the neutrino system where the $Z$ decay contributes, the dependence
on the anomalous parameters is negligible. We can therefore cut on high $VV$
invariant mass without losing sensitivity and thus concentrate on the genuine
VBS topology. This property also removes any need for unitarizing the triple-
production channel, since the SM contribution is guaranteed to respect
perturbative unitarity.

##  6 Results for EFT Operator Coefficients

We now consider the extrapolated EFT as the most straightforward extension of
the SM for VBS at high energy. We use the SM Lagrangian (17) and add the two
dimension-eight operators (18) with their coefficients $F_{S,0}$ and $F_{S,1}$
as free parameters. The calculation is analogous to the LHC case [56]; for the
numerical results, we use the model implementation in the WHIZARD event
generator and the automatic calculation of unitarized tree-level amplitudes,
distributions, and cross sections.

At low energy, the free parameters $F_{S,0}$ and $F_{S,1}$ are identified as
expansion parameters in the EFT, within the range of validity of the
latter.333In the ATLAS analysis of VBS at the LHC [6], the notation
$(\alpha_{4},\alpha_{5})$ was adopted instead, borrowed from the no-Higgs EFT
[45], although the theory model included the light Higgs and unitarization,
i.e., worked with the extrapolated EFT model of the present paper. For the
relation of coefficients, cf. [43]. Where we extrapolate the EFT to high
energies beyond its validity range, we apply T-matrix unitarization to the
calculated amplitude. The unitarization correction will set in where the
extrapolated amplitude becomes strong. In effect, we obtain a smooth
transition to high energy asymptotics with saturation of unitarity in all
partial waves. This is a two-parameter model for a structureless strongly
interacting continuum.

This expectation is confirmed by the numerical results which we display in
Figs. 4, 5, 6. In the plotted distributions, we choose exemplary values for
the model parameters. As in the SM results listed above, all distributions are
shown for unpolarized, structureless $e^{+}e^{-}$ beams.

The result of this calculation can be expressed in terms of exclusion contours
in a two-dimensional parameter space, centered on the SM as reference point.
The exclusion contours are based on the hypothesis that no deviations from the
SM are observed in an experiment. Given that our estimates follow from a
simple cut-based analysis, we find it sufficient to calculate a small number
of parameter points and suitably interpolate. In Fig. 7, we show the results
for the sensitivity to $F_{S,0}$ and $F_{S,1}$. In addition to the exclusion
contours that we obtain for the $W^{+}W^{-}$ and $ZZ$ final states,
respectively, we indicate a 90 % exclusion limit that would be deduced from
combining both channels. Comparing the three selected collider energies, we
conclude that increasing the energy from $1\;\text{TeV}$ to $3\;\text{TeV}$
improves the sensitivity by roughly one order of magnitude, ultimately $\Delta
F_{S,0/1}\sim 5\,\text{TeV}^{-4}$. These results may be compared to the run-I
LHC limit on the same parameters, as obtained by ATLAS [7], $\Delta
F_{S,0/1}\sim 500\,\text{TeV}^{-4}$ (cf. [46] for the conversion of exclusion
limits).

In Fig. 8, we repeat the same analysis for an assumed beam polarization of 80
% ($e^{-}$), 0 % ($e^{+}$). The polarization effect enhances the signal more
than the background and thus improves the sensitivity by another factor of
$1.5$. Finally, Fig. 9 displays the same 90 % exclusion contours with
polarization for all three energy values in a common plot, for convenience of
the reader.

We deliberately have included only the two channels $W^{+}W^{-}$ and $ZZ$.
Additional information can be gained by evaluating the other possible final
states. However, these channels suffer from a larger fraction of
$\gamma$-induced background and add independent information only if we relax
the custodial-symmetry assumption, such that $W$ and $Z$ states are no longer
related. For the current analysis, additional channels are of minor
importance.

Figure 4: Differential cross sections depending on the transverse momentum of
the $W$ (left plots) and the $Z$ boson pair (right plots) at center-of-mass
energies of $1\;\text{TeV}$ (upper plots), $1.4\;\text{TeV}$ (middle plots)
and $3\;\text{TeV}$ (lower plots). The solid lines show the signal process
$\bar{\nu}\nu W^{+}W^{-}(ZZ)$ with SM values $F_{S,0}=F_{S,1}=0$. The red/blue
lines indicate the signal process with non SM value
$F_{S,0,1}=25/50\;\text{TeV}\,^{-4}$ (dashed lines: naive EFT results, solid
lines: unitarized results). In addition, the two SM background processes
$W^{+}W^{-}e^{+}e^{-}$ and $W^{\pm}Ze^{\mp}\nu$ (with 13.6% misidentification
probability) are also plotted. The shaded area is removed by the cut on the
$W(Z)$ boson system. All other cuts have been applied as described. No
detection efficiencies are included.

Figure 5: Differential cross sections depending on the invariant mass of the
$W$ (left plots) and the $Z$ boson pair (right plots) at center-of-mass
energies of $1\;\text{TeV}$ (upper plots), $1.4\;\text{TeV}$ (middle plots)
and $3\;\text{TeV}$ (lower plots). The solid lines show the signal process
$\bar{\nu}\nu W^{+}W^{-}(ZZ)$ with SM values $F_{S,0}=F_{S,1}=0$. The red/blue
lines indicate the signal process with non SM value
$F_{S,0,1}=25/50\;\text{TeV}\,^{-4}$ (dashed lines: naive EFT results, solid
lines: unitarized results). In addition, the two SM background processes
$W^{+}W^{-}e^{+}e^{-}$ and $W^{\pm}Ze^{\mp}\nu$ (with 13.6% misidentification
probability) are also plotted. The shaded area is removed by the cut on the
$W(Z)$ boson system. All other cuts have been applied as described. No
detection efficiencies are included.

Figure 6: Differential cross sections depending on the recoil mass of the $W$
(left plots) and the $Z$ boson pair (right plots) at center-of-mass energies
of $1\;\text{TeV}$ (upper plots), $1.4\;\text{TeV}$ (middle plots) and
$3\;\text{TeV}$ (lower plots). The solid lines show the signal process
$\bar{\nu}\nu W^{+}W^{-}(ZZ)$. The red lines show the background process
$e^{+}e^{-}\to W^{+}W^{-}(ZZ)Z\to W^{+}W^{-}(ZZ)\nu\bar{\nu}$, where the two
neutrinos are generated through a decaying $Z$ boson. The shaded area is
removed by the cut on the neutrino system. All other cuts have been applied as
described above. No detection efficiencies are included.

Figure 7: $\pm 1\sigma$ exclusion contours in the $F_{S,0}/F_{S,1}$ plane for
the two signal processes $e^{+}e^{-}\to\bar{\nu}\nu W^{+}W^{-}$ and
$e^{+}e^{-}\to\bar{\nu}\nu ZZ$ including all background processes based on the
assumption $F_{S,0}=F_{S,1}=0$. The $e^{-}(e^{+})$ beam is unpolarized at
energies of $1.0\;\text{TeV}$ (upper left plot), $1.4\;\text{TeV}$ (upper
right plot) and $3\;\text{TeV}$ (lower plot). The corresponding integrated
luminosities are $5\;\text{ab}^{-1},\ 1.5\;\text{ab}^{-1}$ and
$2\;\text{ab}^{-1}$, respectively. All cuts have been applied and detection
efficiencies are included. The thick line indicates the 90% exclusion
sensitivities obtained by the combination of the two signal channels. All
cross sections are unitarized.

Figure 8: $\pm 1\sigma$ exclusion contours in the $F_{S,0}/F_{S,1}$ plane for
the two signal processes $e^{+}e^{-}\to\bar{\nu}\nu W^{+}W^{-}$ and
$e^{+}e^{-}\to\bar{\nu}\nu ZZ$ including all background processes based on the
assumption $F_{S,0}=F_{S,1}=0$. The $e^{-}(e^{+})$ beam is polarized at a
degree of 80%(0%) at energies of $1.0\;\text{TeV}$ (upper left plot),
$1.4\;\text{TeV}$ (upper right plot) and $3\;\text{TeV}$ (lower plot). The
corresponding integrated luminosities are $5\;\text{ab}^{-1},\
1.5\;\text{ab}^{-1}$ and $2\;\text{ab}^{-1}$, respectively. All cuts have been
applied and detection efficiencies are included. The thick line indicates the
90% exclusion sensitivities obtained by the combination of the two signal
channels. All cross sections are unitarized. Figure 9: 90 % exclusion
sensitivities for polarized (solid) and unpolarized (dashed) particle beams at
energies of $\sqrt{s}=1\text{ (black)},\ 1.4\text{ (blue)},\
3\,\text{TeV}\text{ (red)}$ combined, assuming integrated luminosities of
$5\;\text{ab}^{-1},\ 1.5\;\text{ab}^{-1}$ and $2\;\text{ab}^{-1}$,
respectively.

The sensitivity range for the anomalous coupling is of the order
$1/\Lambda_{\text{eff}}^{4}$, where the effective scale $\Lambda_{\text{eff}}$
is given by the $W/Z$ pair energy where the differential distribution is
maximal, cf. Fig. 5. This is the region where the anomalous couplings have the
strongest impact — at low energies, their effect is naturally suppressed,
while at high energies, unitarity becomes saturated and the sensitivity
disappears again. Clearly, the measurement is limited by statistics.

For the purpose of this study, we rely only on the total cross section within
a fiducial phase space which is isolated by the cut strategy as described
above. Further details can be deduced from the vector-boson pair invariant
mass distribution, as displayed in the figures. The availability of this
important observable, in form of the total hadronic energy and momentum, is a
great advantage of the clean lepton-collider environment, as compared to the
situation at a hadron collider.

A complete experimental analysis, which is beyond the scope of the present
paper, should take the simulation results and apply a proper fitting procedure
that takes into account all available information. Such an analysis of VBS
data would exploit a complete set of observables. In particular, it will be
advantageous to resolve the decay products of the vector bosons into
individual jets – quarks in the language of the partonic elementary process –
and take into account their angular and energy distributions. For illustration
of the added value, we have generated WHIZARD event samples for the complete
exclusive process $e^{+}e^{-}\to\bar{\nu}\nu+4j$ with all possible Feynman
graphs included, summed over neutrino and quark flavors, for the SM and for
some nonzero values of the EFT operator coefficient $F_{S,0}$, Fig. 10.

Figure 10: Differential cross sections of the process
$e^{+}e^{-}\bar{\nu}\to\nu+4j$ at center-of-mass energies of $1.4\;\text{TeV}$
(left plot) and $3\;\text{TeV}$ (right plot) depending on the jet pairs
$|cos(\theta^{*})|$ at different values of $F_{S,0}$. All cuts have been
applied as described above. No detection efficiencies are included and all
cross sections are unitarized.

In this figure, we show the distribution in the polar angle $\theta^{*}$
between the final state jets in the rest frame of the parent (off-shell)
vector boson. This cut is applied at Monte Carlo truth level to both jet pairs
including all combinatorics. Expanding on this result, to enhance the vector
boson scattering signal further above the background, the following cut on the
angle $\theta^{*}$ could be used (as before: first number applies for
$3\;\text{TeV}$ and the number in brackets for $1.4\;\text{TeV}$):

$0.2(0.4)<|\cos(\theta^{*})|<0.75$ (21)

A more sophisticated analysis would exploit the complete information from this
distribution and add in any further observable that provides discriminating
power.

##  7 Results for Resonances

Besides the extrapolated EFT of the previous section, we now consider a second
set of models for VBS at high energy. Starting again from the Lagrangian (17),
we add a field with high mass, definite $SU(2)_{L}\times SU(2)_{R}$ quantum
numbers and with a minimal coupling to a Higgs-Goldstone current in the
Lagrangian, as a possible resonance. For concreteness, we adopt resonance
quantum numbers and parameters that we did study for the LHC in Ref. [56], and
specifically select three models, namely an isoscalar-scalar $\sigma$, an
isotensor-scalar $\phi$, and an isoscalar-tensor $f$. For each model, we
introduce the resonance mass and the coupling to the Higgs current as two
independent new parameters. Ignoring further decay channels, the coupling and
mass also determine the width of the resonance.

The Lagrangians for the three fields $\sigma$, $f$, $\phi$ are

$\displaystyle\mathcal{L}_{\sigma}=$
$\displaystyle\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\sigma
J_{\sigma}\,$ (22a) $\displaystyle\mathcal{L}_{\phi}=$
$\displaystyle\frac{1}{2}\sum_{i=s,v,t}\operatorname{tr}\left[\partial_{\mu}\Phi_{i}\partial^{\mu}\Phi_{i}-m_{\Phi}^{2}\Phi_{i}^{2}\right]+\operatorname{tr}\left[\left(\Phi_{t}+\frac{1}{2}\Phi_{v}-\frac{2}{5}\Phi_{s}\right)J_{\phi}\right]\,,$
(22b) $\displaystyle\mathcal{L}_{f}=$
$\displaystyle\frac{1}{2}\partial_{\alpha}f_{\mu\nu}\partial^{\alpha}f^{\mu\nu}-\frac{1}{2}m^{2}f_{\mu\nu}f^{\mu\nu}$
$\displaystyle\;-\partial^{\alpha}f_{\alpha\mu}\partial_{\beta}f^{\beta\mu}-f^{\alpha}_{\;\alpha}\partial^{\mu}\partial^{\nu}f_{\mu\nu}-\frac{1}{2}\partial_{\alpha}f^{\mu}_{\;\mu}\partial^{\alpha}f^{\nu}_{\;\nu}+\frac{1}{2}m^{2}f^{\mu}_{\;\mu}f^{\nu}_{\;\nu}+f_{\mu\nu}J_{f}^{\mu\nu},$
(22c)

where $J_{\sigma},J_{\phi},J_{f}$ are the currents which couple to the new
fields, respectively. We note that the isotensor-scalar $\phi$, defined by its
$SU(2)_{R}\times SU(2)_{L}$ quantum numbers $\mathbf{1}\times\mathbf{1}$,
decomposes after electroweak symmetry breaking into an isotensor $\Phi_{t}$,
an isovector $\Phi_{v}$ and isoscalar $\Phi_{s}$ transforming under custodial
$SU(2)$.

The currents that interact with the resonances are given by

$\displaystyle J_{\sigma}$ $\displaystyle=F_{\sigma}$
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\mu}\mathbf{H}\right]\,,$
(23a) $\displaystyle J_{\phi}$ $\displaystyle=F_{\phi}$
$\displaystyle\left(\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\otimes\mathbf{D}^{\mu}\mathbf{H}+\frac{1}{8}\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\mu}\mathbf{H}\right]\right)\tau^{a}\otimes\tau^{a}\,,$
(23b) $\displaystyle J^{\mu\nu}_{f}$ $\displaystyle=F_{f}$
$\displaystyle\left(\operatorname{tr}\left[\left(\mathbf{D}^{\mu}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\nu}\mathbf{H}\right]-\frac{c_{f}}{4}g^{\mu\nu}\operatorname{tr}\left[\left(\mathbf{D}_{\rho}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\rho}\mathbf{H}\right]\right)\,.$
(23c)

For further details, cf. [56].

At low energy, each of these three distinct models reduces to a one-parameter
effective theory, since a single combination of mass and coupling parameters
enters into effective values of $F_{S,0}$ and $F_{S,1}$. These models
therefore provide high-energy extensions of the generic low-energy EFT, as
alternatives to the straightforward extrapolation of the preceding section.
The resonance models exhibit more structure than the simple extrapolation. All
models require unitarization, which we again implement using the T-matrix
framework, since the interaction operators introduce terms that rise with
energy.

If the effect of the new states is to be sizable and thus observable, the
interactions should be rather strong, so generically we do not expect a
renormalizable model with tree-level unitary asymptotics. Nevertheless, weakly
interacting models such as a UV-complete Higgs-singlet model are included in
this model space. For a renormalizable model, any terms that apparently rise
with energy would cancel against higher-order contributions. From a
phenomenological perspective, renormalizable models are exceptional points in
a larger parameter space that we cover by the above definitions.

Figure 11: Differential cross sections including a weakly coupled isoscalar
scalar resonance (${m_{\sigma}=800\;\text{GeV}}$,
$F_{\sigma}=4.0\;\text{TeV}^{-1}$, $\Gamma_{\sigma}=80\;\text{GeV}$) depending
on the invariant mass of the vector boson pair at center-of-mass energies of
$\sqrt{s}=1.4\;\text{TeV}$ (upper plots) and $\sqrt{s}=3\;\text{TeV}$ (lower
plots). Plots on the left show the process $e^{+}e^{-}\to\bar{\nu}\nu
W^{+}W^{-}$, plots on the right $e^{+}e^{-}\to\bar{\nu}\nu ZZ$. Blue line:
isoscalar scalar resonance, red line: matched EFT results ($F_{S,0}=0$,
$F_{S,1}=12.3\;\text{TeV}^{-4}$). Solid line: unitarized results, dashed line:
naive results.

In Fig. 12, we display the vector-boson pair invariant mass distribution for
the SM with an additional isoscalar scalar resonance $\sigma$, equivalent to
an extra Higgs-like singlet boson. The plots show both the $W^{+}W^{-}$ final
state (left column) and the $ZZ$ final state (right column). We have chosen a
moderately high mass of $M_{\sigma}=800\;\text{GeV}$ and a rather small width
of $\Gamma_{\sigma}=80\;\text{GeV}$ (blue curve). The distribution, which for
the full process translates into the invariant mass of the hadronic (four-jet)
system, shows an unambiguous peak at the resonance mass that is
distinguishable in shape from the SM background (black), given sufficient
luminosity. The peak is more pronounced for $3\;\text{TeV}$ collider energy,
but also clearly visible for $1.4\;\text{TeV}$. It is evident that for this
choice of parameters, the formally correct EFT expansion (red curve) does not
aproximate the actual model behavior at all. Unitarization does not play a
significant role except for the naively extrapolated EFT (dashed red) which
overshoots the unitarized curves (solid), and thus the unitarity bounds, by a
substantial amount.

Figure 12: Differential cross sections including a low lying isoscalar scalar
resonance (${m_{\sigma}=650\;\text{GeV}}$, $F_{\sigma}=9.8\;\text{TeV}^{-1}$,
$\Gamma_{\sigma}=260\;\text{GeV}$) depending on the invariant mass of the
vector boson pair at center-of-mass energies of $\sqrt{s}=1.4\;\text{TeV}$
(upper plots) and $\sqrt{s}=3\;\text{TeV}$ (lower plots). Plots on the left
show the process $e^{+}e^{-}\to\bar{\nu}\nu W^{+}W^{-}$, plots on the right
$e^{+}e^{-}\to\bar{\nu}\nu ZZ$. Blue line: isoscalar scalar resonance, red
line: matched EFT results ($F_{S,0}=0$, $F_{S,1}=112.6\;\text{TeV}^{-4}$).
Solid line: unitarized results, dashed line: naive results.

The plots in Fig. 12 depict a somewhat lighter isoscalar-scalar resonance
($M_{\sigma}=650\;\text{GeV}$) with a larger width
($\Gamma_{\sigma}=260\;\text{GeV}$). In this case, the resonance shape follows
the shape of the extrapolated EFT but leads to at significantly larger peak
cross section. Since the equivalent EFT parameter, matching to the slope of
the resonance curve below threshold, is twice as large as in the previous set
of plots, the unphysical behavior is evident that we would get without
unitarization (dashed curves). This applies not just to the EFT approximation
(red), but also to the resonance curves (blue).

Figure 13: Differential cross sections including an isoscalar tensor resonance
(${m_{f}=1000\;\text{GeV}}$, $F_{f}=17.4\;\text{TeV}^{-1}$,
$\Gamma_{f}=100\;\text{GeV}$) depending on the invariant mass of the vector
boson pair at center-of-mass energies of $\sqrt{s}=1.4\;\text{TeV}$ (upper
plots) and $\sqrt{s}=3\;\text{TeV}$ (lower plots). Plots on the left show the
process $e^{+}e^{-}\to\bar{\nu}\nu W^{+}W^{-}$, plots on the right
$e^{+}e^{-}\to\bar{\nu}\nu ZZ$. Blue line: isoscalar tensor resonance, red
line: matched EFT results ($F_{S,0}=150.8\;\text{TeV}^{-4}$,
$F_{S,1}=-50.3\;\text{TeV}^{-4}$). Solid line: unitarized results, dashed
line: naive results.

In Fig 13, we consider a isoscalar tensor resonance with $M_{f}=1\;\text{TeV}$
and $\Gamma_{f}=100\;\text{GeV}$, which produces a rather narrow peak in all
distributions. With a collider energy of $3\;\text{TeV}$, we observe the
necessity for unitarization beyond the mass peak (in the $WW$ final state),
caused by the dimensionality of the effective tensor-scalar interaction. As
well as in all other cases, if we had a UV-complete model at hand, we would
expect any variation of the resonance-model prediction (blue) in this range:
further resonances, a featureless continuum, or suppression that accommodates
the emergence of further inelastic channels. However, neither of these
scenarios could produce a unitarity-violating result like the naive blue-
dashed line, so the unitarized model prediction serves as a conservative
estimate of the asymptotic shape.

Figure 14: Differential cross sections including a low lying isotensor scalar
resonance (${m_{\phi}=650\;\text{GeV}}$, $F_{\phi}=19.5\;\text{TeV}^{-1}$,
$\Gamma_{\phi}=260\;\text{GeV}$) depending on the invariant mass of the vector
boson pair at center-of-mass energies of $\sqrt{s}=1.4\;\text{TeV}$ (upper
plots) and $\sqrt{s}=3\;\text{TeV}$ (lower plots). Plots on the left show the
process $e^{+}e^{-}\to\bar{\nu}\nu W^{+}W^{-}$, plots on the right
$e^{+}e^{-}\to\bar{\nu}\nu ZZ$. Blue line: isotensor scalar resonance, red
line: matched EFT results
($F_{S,0}=450.5\;\text{TeV}^{-4}$,$F_{S,1}=-112.6\;\text{TeV}^{-4}$). Solid
line: unitarized results, dashed line: naive results.

The final parameter set in Fig. 14, an isotensor-scalar multiplet $\phi$,
illustrates a possible strongly interacting multi-Higgs scenario. The broad
resonance, actually a combination of resonance exchange in all isospin
channels, is indistinguishable from an arbitrary continuum. It is remarkable
that the EFT approximation follows the shape of the resonance model in the
$WW$ final state but fails completely for the $ZZ$ final state. As before, the
naive extrapolations (dashed) overshoot the unitarized models (solid) by a
large amount.

##  8 Conclusions

We have performed a new study of the capability of a high-energy lepton
collider (such as CLIC or an upgraded ILC) to measure quasi-elastic vector-
boson scattering, as a dedicated probe of the Higgs sector. For realistic
luminosity-energy combinations, we cannot restrict the investigation to a pure
effective field theory (EFT), but have to take into account strong
interactions or resonant behavior. Specifically, we have considered a
straightforwardly extrapolated EFT Lagrangian and related this to alternative
models where the anomalous effects develop into resonances within the
kinematical range.

Our phenomenological scenarios all reduce to the Standard Model with leading
higher-dimensional corrections at the low-energy threshold for vector boson
scattering processes, but run into unitarity saturation at high energy or on a
resonance peak. The T-matrix unitarization framework lets us handle this issue
by a universal algorithm and simultaneously allows us to embed the models in a
full off-shell calculation and Monte-Carlo simulation. For the purpose of
working with fully exclusive event samples, all models have been implemented
in the public version of the WHIZARD Monte-Carlo event generator.

The results yield an overview over the experimental reach of the CLIC and ILC
colliders in the various scenarios. Using optimized cuts for longitudinal
vector boson scattering, we generically obtain a sensitivity on the
extrapolated EFT parameters $F_{S,0}$ and $F_{S,1}$ of $O(1/\Lambda^{4})$,
where $\Lambda$ indicates the location of the maximum of the unitarized cross
section, in terms of the vector-boson pair invariant mass as the relevant
energy scale. Regarding resonance models, we have restricted ourselves to a
few exemplary parameter points in this exploratory study, which should be
representative of the phenomenology that can be expected. For precise and
systematic parameter determinations, the specific models should be subject to
an experimental fitting procedure and analysis which takes into account
complete event information and weighs the distributions according to their
sensitivity to the specific model in question.

The expected sensitivity of the CLIC collider with $3\,\text{TeV}$, as
expressed for the parameters of the extrapolated EFT as the simplest model,
improves over the current LHC limits by two orders of magnitude. Clearly, the
CLIC expectation has to be compared to the ultimate precision achievable by
the LHC experiments. A detailed comparison of sensitivities would require
applying the available analysis techniques to a full-simulation prediction for
either collider, which is beyond the scope of the present paper. However, the
$e^{+}e^{-}$ environment allows for the detection of well-defined final
states, hadronic decays, and a direct measurement of the most relevant
distribution – the vector-boson pair invariant mass – and thus remains the
preferred setup for a comprehensive study of VBS in the TeV energy range.

## Acknowledgments

We would like to thank Lucie Linssen, Philipp Roloff and Marcel Vos for
enlightening discussions on the CLIC project. MS acknowledges the support of
BMBF Verbundforschung (HEP Theory).

##  Appendix A Dimension-8 Operators that Affect VBS

The following list of dimension-8 operators includes all leading interactions
that modify the SM form of VBS interactions. In the current paper, we restrict
the investigation to the first set of operators which describes genuine
Goldstone-Higgs self-interactions, which for completeness we repeat here from
18:

$\displaystyle\mathcal{L}_{S,0}$ $\displaystyle=$ $\displaystyle F_{S,0}\ $
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\mathbf{D}_{\nu}\mathbf{H}\right]\operatorname{tr}\left[\left(\mathbf{D}^{\mu}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\nu}\mathbf{H}\right],$
(24a) $\displaystyle\mathcal{L}_{S,1}$ $\displaystyle=$ $\displaystyle
F_{S,1}\ $
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\mu}\mathbf{H}\right]\operatorname{tr}\left[\left(\mathbf{D}_{\nu}\mathbf{H}\right)^{\dagger}\mathbf{D}^{\nu}\mathbf{H}\right].$
(24b)

Exchanging two covariant derivatives with field strength tensors, further
possibilities arise to construct linearly independent operators

$\displaystyle\mathcal{L}_{M,0}$ $\displaystyle=$ $\displaystyle-g^{2}$
$\displaystyle F_{M,0}$
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\left(\mathbf{D}^{\mu}\mathbf{H}\right)\right]\operatorname{tr}\left[\mathbf{W}_{\nu\rho}\mathbf{W}^{\nu\rho}\right],$
(25a) $\displaystyle\mathcal{L}_{M,1}$ $\displaystyle=$ $\displaystyle-g^{2}$
$\displaystyle F_{M,1}$
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\left(\mathbf{D}^{\rho}\mathbf{H}\right)\right]\operatorname{tr}\left[\mathbf{W}_{\nu\rho}\mathbf{W}^{\nu\mu}\right],$
(25b) $\displaystyle\mathcal{L}_{M,2}$ $\displaystyle=$
$\displaystyle-{g^{\prime}}^{2}$ $\displaystyle F_{M,2}$
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\left(\mathbf{D}^{\mu}\mathbf{H}\right)\right]\operatorname{tr}\left[\mathbf{B}_{\nu\rho}\mathbf{B}^{\nu\rho}\right],$
(25c) $\displaystyle\mathcal{L}_{M,3}$ $\displaystyle=$
$\displaystyle-{g^{\prime}}^{2}$ $\displaystyle F_{M,3}$
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\left(\mathbf{D}^{\rho}\mathbf{H}\right)\right]\operatorname{tr}\left[\mathbf{B}_{\nu\rho}\mathbf{B}^{\nu\mu}\right],$
(25d) $\displaystyle\mathcal{L}_{M,4}$ $\displaystyle=$ $\displaystyle-
gg^{\prime}$ $\displaystyle F_{M,4}$
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\mathbf{W}_{\nu\rho}\left(\mathbf{D}^{\mu}\mathbf{H}\right)\mathbf{B}^{\nu\rho}\right],$
(25e) $\displaystyle\mathcal{L}_{M,5}$ $\displaystyle=$ $\displaystyle-
gg^{\prime}$ $\displaystyle F_{M,5}$
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\mathbf{W}_{\nu\rho}\left(\mathbf{D}^{\rho}\mathbf{H}\right)\mathbf{B}^{\nu\mu}\right],$
(25f) $\displaystyle\mathcal{L}_{M,7}$ $\displaystyle=$ $\displaystyle-g^{2}$
$\displaystyle F_{M,7}$
$\displaystyle\operatorname{tr}\left[\left(\mathbf{D}_{\mu}\mathbf{H}\right)^{\dagger}\mathbf{W}_{\nu\rho}\mathbf{W}^{\nu\mu}\left(\mathbf{D}^{\rho}\mathbf{H}\right)\right].$
(25g)

Here, we kept the numbering analogue to the linear Higgs doublet
representation in [43] for future comparisons, where some linear dependent
operators of [41] are already omitted.

Operators affecting only the gauge bosons consist of four electroweak field
strength tensors

$\displaystyle\mathcal{L}_{T,0}$ $\displaystyle=$ $\displaystyle g^{4}$
$\displaystyle F_{T,0}$
$\displaystyle\operatorname{tr}\left[\mathbf{W}_{\mu\nu}\mathbf{W}^{\mu\nu}\right]\operatorname{tr}\left[\mathbf{W}_{\alpha\beta}\mathbf{W}^{\alpha\beta}\right],$
(26a) $\displaystyle\mathcal{L}_{T,1}$ $\displaystyle=$ $\displaystyle g^{4}$
$\displaystyle F_{T,1}$
$\displaystyle\operatorname{tr}\left[\mathbf{W}_{\alpha\nu}\mathbf{W}^{\mu\beta}\right]\operatorname{tr}\left[\mathbf{W}_{\mu\beta}\mathbf{W}^{\alpha\nu}\right],$
(26b) $\displaystyle\mathcal{L}_{T,2}$ $\displaystyle=$ $\displaystyle g^{4}$
$\displaystyle F_{T,2}$
$\displaystyle\operatorname{tr}\left[\mathbf{W}_{\alpha\mu}\mathbf{W}^{\mu\beta}\right]\operatorname{tr}\left[\mathbf{W}_{\beta\nu}\mathbf{W}^{\nu\alpha}\right],$
(26c) $\displaystyle\mathcal{L}_{T,5}$ $\displaystyle=$ $\displaystyle g^{4}$
$\displaystyle F_{T,5}$
$\displaystyle\operatorname{tr}\left[\mathbf{W}_{\mu\nu}\mathbf{W}^{\mu\nu}\right]\operatorname{tr}\left[\mathbf{B}_{\alpha\beta}\mathbf{B}^{\alpha\beta}\right],$
(26d) $\displaystyle\mathcal{L}_{T,6}$ $\displaystyle=$ $\displaystyle g^{4}$
$\displaystyle F_{T,6}$
$\displaystyle\operatorname{tr}\left[\mathbf{W}_{\alpha\nu}\mathbf{W}^{\mu\beta}\right]\operatorname{tr}\left[\mathbf{B}_{\mu\beta}\mathbf{B}^{\alpha\nu}\right],$
(26e) $\displaystyle\mathcal{L}_{T,7}$ $\displaystyle=$ $\displaystyle g^{4}$
$\displaystyle F_{T,7}$
$\displaystyle\operatorname{tr}\left[\mathbf{W}_{\alpha\mu}\mathbf{W}^{\mu\beta}\right]\operatorname{tr}\left[\mathbf{B}_{\beta\nu}\mathbf{B}^{\nu\alpha}\right],$
(26f) $\displaystyle\mathcal{L}_{T,8}$ $\displaystyle=$
$\displaystyle{g^{\prime}}^{4}$ $\displaystyle F_{T,8}$
$\displaystyle\operatorname{tr}\left[\mathbf{B}_{\mu\nu}\mathbf{B}^{\mu\nu}\right]\operatorname{tr}\left[\mathbf{B}_{\alpha\beta}\mathbf{B}^{\alpha\beta}\right],$
(26g) $\displaystyle\mathcal{L}_{T,9}$ $\displaystyle=$
$\displaystyle{g^{\prime}}^{4}$ $\displaystyle F_{T,9}$
$\displaystyle\operatorname{tr}\left[\mathbf{B}_{\alpha\mu}\mathbf{B}^{\mu\beta}\right]\operatorname{tr}\left[\mathbf{B}_{\beta\nu}\mathbf{B}^{\nu\alpha}\right].$
(26h)

## References

  * [1] M. S. Chanowitz and M. K. Gaillard, Nucl. Phys. B 261, 379 (1985).
  * [2] G. J. Gounaris, R. Kogerler and H. Neufeld, Phys. Rev. D 34, 3257 (1986).
  * [3] B. W. Lee, C. Quigg and H. B. Thacker, Phys. Rev. Lett. 38, 883 (1977).
  * [4] B. W. Lee, C. Quigg and H. B. Thacker, Phys. Rev. D 16, 1519 (1977).
  * [5] J. M. Cornwall, D. N. Levin and G. Tiktopoulos, Phys. Rev. D 10, 1145 (1974) Erratum: [Phys. Rev. D 11, 972 (1975)].
  * [6] G. Aad et al. [ATLAS Collaboration], Phys. Rev. Lett. 113, no. 14, 141803 (2014) [arXiv:1405.6241 [hep-ex]].
  * [7] The ATLAS collaboration [ATLAS Collaboration], ATLAS-CONF-2014-013.
  * [8] CMS Collaboration [CMS Collaboration], CMS-PAS-FSQ-13-008.
  * [9] E. Accomando et al. [ECFA/DESY LC Physics Working Group Collaboration], Phys. Rept. 299, 1 (1998) [hep-ph/9705442].
  * [10] T. Abe et al. [Linear Collider American Working Group Collaboration], hep-ex/0106057.
  * [11] J. F. Gunion and A. Tofighi-Niaki, Phys. Rev. D 36, 2671 (1987).
  * [12] A. Tofighi-Niaki and J. F. Gunion, Phys. Rev. D 38, 1433 (1988).
  * [13] V. D. Barger, K. m. Cheung, T. Han and R. J. N. Phillips, Phys. Rev. D 52, 3815 (1995) [hep-ph/9501379].
  * [14] T. Han, H. J. He and C. P. Yuan, Phys. Lett. B 422, 294 (1998) [hep-ph/9711429].
  * [15] D. Dominici, Riv. Nuovo Cim. 20N11, 1 (1997) [Riv. Nuovo Cim. 20, 1 (1997)] [hep-ph/9711385].
  * [16] E. Boos, H. J. He, W. Kilian, A. Pukhov, C. P. Yuan and P. M. Zerwas, Phys. Rev. D 57 (1998) 1553 [hep-ph/9708310].
  * [17] E. Boos, H. J. He, W. Kilian, A. Pukhov, C. P. Yuan and P. M. Zerwas, Phys. Rev. D 61 (2000) 077901 [hep-ph/9908409].
  * [18] R. Chierici, S. Rosati and M. Kobel, LC-PHSM-2001-038.
  * [19] S. Rosati, PhD thesis.
  * [20] M. Beyer, W. Kilian, P. Krstonosic, K. Mönig, J. Reuter, E. Schmidt and H. Schröder, Eur. Phys. J. C 48, 353 (2006) [hep-ph/0604048].
  * [21] S. Liebler, G. Moortgat-Pick and G. Weiglein, JHEP 1506, 093 (2015) [arXiv:1502.07970 [hep-ph]].
  * [22] A. Denner, S. Dittmaier and T. Hahn, Phys. Rev. D 56, 117 (1997) [hep-ph/9612390].
  * [23] A. Denner and T. Hahn, Nucl. Phys. B 525, 27 (1998) [hep-ph/9711302].
  * [24] E. Accomando, A. Denner and S. Pozzorini, JHEP 0703, 078 (2007) [hep-ph/0611289].
  * [25] L. Linssen, A. Miyamoto, M. Stanitzki and H. Weerts, arXiv:1202.5940 [physics.ins-det].
  * [26] P. Lebrun et al., arXiv:1209.2543 [physics.ins-det].
  * [27] M. Aicheler et al., CERN-2012-007, SLAC-R-985, KEK-Report-2012-1, PSI-12-01, JAI-2012-001.
  * [28] H. Baer et al., arXiv:1306.6352 [hep-ph].
  * [29] T. Behnke et al., arXiv:1306.6329 [physics.ins-det].
  * [30] T. Appelquist and J. Carazzone, Phys. Rev. D 11, 2856 (1975).
  * [31] S. Weinberg, Physica A 96, 327 (1979).
  * [32] H. Georgi, Ann. Rev. Nucl. Part. Sci. 43, 209 (1993).
  * [33] R. Contino, C. Grojean, D. Pappadopulo, R. Rattazzi and A. Thamm, JHEP 1402, 006 (2014) [arXiv:1309.7038 [hep-ph]].
  * [34] M. J. G. Veltman, Nucl. Phys. B 123, 89 (1977).
  * [35] P. Sikivie, L. Susskind, M. B. Voloshin and V. I. Zakharov, Nucl. Phys. B 173, 189 (1980).
  * [36] W. Kilian, Springer Tracts Mod. Phys. 198, 1 (2003).
  * [37] W. Kilian, T. Ohl, J. Reuter and M. Sekulla, Phys. Rev. D 91 (2015) 096007 [arXiv:1408.6207 [hep-ph]].
  * [38] W. Buchmüller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B 268 (1986) 621.
  * [39] K. Hagiwara, S. Ishihara, R. Szalapski and D. Zeppenfeld, Low-energy constraints on electroweak three gauge boson couplings, Phys. Lett. B 283 (1992) 353.
  * [40] K. Hagiwara, S. Ishihara, R. Szalapski and D. Zeppenfeld, Low-energy effects of new interactions in the electroweak boson sector, Phys. Rev. D 48 (1993) 2182.
  * [41] O. J. P. Eboli, M. C. Gonzalez-Garcia and J. K. Mizukoshi, Phys. Rev. D 74, 073005 (2006) [hep-ph/0606118].
  * [42] B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 1010 (2010) 085 [arXiv:1008.4884 [hep-ph]].
  * [43] M. Baak et al., arXiv:1310.6708 [hep-ph].
  * [44] A. C. Longhitano, Nucl. Phys. B 188, 118 (1981).
  * [45] T. Appelquist and G. H. Wu, Phys. Rev. D 48, 3235 (1993) [hep-ph/9304240].
  * [46] M. Sekulla “Anomalous Couplings, Resonances and Unitarity in Vector Boson Scattering,” PhD thesis, University of Siegen (2015)
  * [47] A. Alboteanu, W. Kilian and J. Reuter, JHEP 0811, 010 (2008) [arXiv:0806.4145 [hep-ph]].
  * [48] W. Kilian, T. Ohl and J. Reuter, WHIZARD: Simulating Multi-Particle Processes at LHC and ILC, Eur. Phys. J. C 71, 1742 (2011) [arXiv:0708.4233 [hep-ph]].
  * [49] M. Moretti, T. Ohl and J. Reuter, O’Mega: An Optimizing matrix element generator, In *2nd ECFA/DESY Study 1998-2001* 1981-2009 [hep-ph/0102195].
  * [50] W. Kilian, J. Reuter, S. Schmidt and D. Wiesler, JHEP 1204, 013 (2012) [arXiv:1112.1039 [hep-ph]].
  * [51] N. D. Christensen, C. Duhr, B. Fuks, J. Reuter and C. Speckner, Eur. Phys. J. C 72, 1990 (2012) [arXiv:1010.3251 [hep-ph]].
  * [52] W. Kilian, T. Ohl, J. Reuter and C. Speckner, JHEP 1210, 022 (2012) [arXiv:1206.3700 [hep-ph]].
  * [53] B. Chokoufe Nejad, W. Kilian, J. Reuter and C. Weiss, PoS EPS -HEP2015, 317 (2015) [arXiv:1510.02739 [hep-ph]]; C. Weiss, B. Chokoufe Nejad, W. Kilian and J. Reuter, PoS EPS -HEP2015, 466 (2015) [arXiv:1510.02666 [hep-ph]]; N. Greiner, A. Guffanti, T. Reiter and J. Reuter, Phys. Rev. Lett. 107, 102002 (2011) [arXiv:1105.3624 [hep-ph]]; T. Binoth, N. Greiner, A. Guffanti, J. Reuter, J.-P. Guillet and T. Reiter, Phys. Lett. B 685, 293 (2010) [arXiv:0910.4379 [hep-ph]]; W. Kilian, J. Reuter and T. Robens, Eur. Phys. J. C 48, 389 (2006) [hep-ph/0607127].
  * [54] M. Idzik [FCAL Collaboration], Acta Phys. Polon. B 46, no. 7, 1297 (2015).
  * [55] J. S. Marshall, A. Münnich and M. A. Thomson, Nucl. Instrum. Meth. A 700, 153 (2013) [arXiv:1209.4039 [physics.ins-det]].
  * [56] W. Kilian, T. Ohl, J. Reuter and M. Sekulla, Phys. Rev. D 93, no. 3, 036004 (2016) [arXiv:1511.00022 [hep-ph]].