# The Curious Case of End Token: A Zero-Shot Disentangled Image Editing using
CLIP

Hidir Yesiltepe Yusuf Dalva Pinar Yanardag
Virginia Tech
{hidir, ydalva<EMAIL_ADDRESS>
###### Abstract

Diffusion models have become prominent in creating high-quality images.
However, unlike GAN models celebrated for their ability to edit images in a
disentangled manner, diffusion-based text-to-image models struggle to achieve
the same level of precise attribute manipulation without compromising image
coherence. In this paper, CLIP which is often used in popular text-to-image
diffusion models such as Stable Diffusion is capable of performing
disentangled editing in a zero-shot manner. Through both qualitative and
quantitative comparisons with state-of-the-art editing methods, we show that
our approach yields competitive results. This insight may open opportunities
for applying this method to various tasks, including image and video editing,
providing a lightweight and efficient approach for disentangled editing.

---

Figure 1: Disentagled image editing using <EOS> token. Original images are
displayed on the left side, and the edited versions are on the right. All
edits are conducted using the <EOS> token related to the respective attribute,
such as the <EOS> token of ‘A woman with eyeglasses’ or ‘A man with mustache’.

##  1 Introduction

Figure 2: Given a source embedding $s$ of a text prompt such ‘A woman’ and a
target embedding $g$ such as ‘A person with an eyeglass’, we would like to
modify the source embedding $s$ according to $g$ to reflect the corresponding
change by replacing <EOS> token of $g$ with source embedding $s$.

Denoising Diffusion Models (DDPMs) [8] and Latent Diffusion Models (LDMs) [13]
have garnered significant interest for their capacity to generate high-
quality, high-resolution images across various domains. They have marked
notable achievements in generative modeling, especially with text-to-image
models such as Stable Diffusion [13].

In generative models, a key aspect of image editing is the disentangled
manipulation of semantics, aiming to alter semantically relevant areas of an
image without impacting other regions [11, 18]. Prior studies have shown that
latent space disentanglement is achieved easily in GANs as compared to
diffusion models. This has led to considerable research efforts focused on
both supervised and unsupervised methods for navigating latent directions in
GANs [19, 6, 14]. However, achieving disentangled editing in diffusion models
poses a significant challenge. Unlike GANs, which have a structured latent
space that naturally lends itself to such disentangled edits, diffusion models
operate on a different principle that doesn’t inherently support disentangled
editing. This is due to the way diffusion models progressively refine images
from noise, which complicates the precise control over specific image
attributes without affecting others. Despite their impressive capabilities in
generating detailed and coherent images, this limitation has been a bottleneck
for using diffusion models in tasks that require fine-grained semantic
modifications.

Several works have been proposed to achieve disentangled editing in diffusion
models, yet they often necessitate expensive procedures such as additional
training or fine-tuning [4, 3, 17]. These approaches, while effective,
significantly increase the computational and time costs associated with
applying diffusion models for image editing tasks. In this paper, we share an
intriguing observation: the CLIP, integral to text-to-image diffusion models,
covertly functions as a zero-shot image editing tool. This revelation opens
the door to leveraging the capabilities of the CLIP model embedded within
diffusion models for image editing, bypassing the need for costly additional
training processes.

##  2 Related work

The utilization of diffusion models for image editing tasks has garnered
growing interest within the field of image generation. A prevalent approach is
to use text prompts to dictate the desired modifications, but this method
often results in entangled edits, where unintended parts of the image are
inadvertently altered. Noteworthy exceptions, such as the research conducted
by [7] and [20], demonstrate more precise editing techniques. For instance,
[20]’s ControlNet employs a conditional diffusion model, which permits users
to alter specific attributes of an image through conditions. Similarly,
studies like [15] manage to retain the original content integrity by finely
tuning the diffusion model to the input image. Moreover, works by [12, 5, 17,
9] introduce methods for accurate input image reconstruction, enabling
content-preserving edits with classifier-free guidance. While these methods
excel in preserving the original appearance during edits, the need for image-
specific optimization limits their practicality for instantaneous editing
applications.

Recent developments have investigated modifications to the denoising steps of
stochastic diffusion models to streamline the editing process. Although these
advancements promise more realistic modifications, crafting an optimal editing
prompt that maintains the realism and fidelity of the edits to the original
image poses a challenge. To address issues of flexibility, some studies, like
[1] and [10], suggest decomposing the editing process into several stages.
Nonetheless, these methods encounter difficulties when applying multiple edits
concurrently, often resulting in compounded effects when various changes are
made to the same image. Recent investigations, such as by [3], have shown
success in applying disentangled edits in large-scale models like Stable
Diffusion. However, the unsupervised nature of these approaches limits the
identification of a comprehensive set of disentangled directions, thus
narrowing their adaptability.

Figure 3: Target <EOS> Guidance Scale Ablation. We investigate the trade-off
between editing quality vs. preservation depending on the target <EOS>
guidance scale hyperparameter. Figure 4: Qualitative comparison. We compare
the image editing capabilities of <EOS> token with state-of-the-art methods
SEGA [1], Ledits++ [2] and Cycle Diffusion [16].

##  3 Methodology

Given a source embedding $s$ of a text prompt such ‘A woman’ and a target
embedding $g$ such as ‘A person with an eyeglass’, we would like to modify the
source embedding $s$ according to $g$ to reflect the corresponding change. Our
method is inspired by a novel observation: <EOS> token in CLIP model is
capable of performing disentangled edits in a zero-shot manner. By leveraging
the fact that CLIP has a decoder-only text encoder and performs a causal
language encoding, we can utilize the <EOS> representation of the target
embedding $g$ to change the context of the source embedding $s$. Thus, we
define $\sigma(s,g)$ for arbitrary text conditions $s$ and $g$ as follows:

$\sigma(s,g)=[s_{\text{{<SOS>}:N}}\mid w\times g_{\texttt{<EOS>}}],$ (1)

in which $s_{\text{[SOS]:N}}\in\mathbb{R}^{d\times(N+1)}$ where <SOS>
represents start of the sentence token,
$g_{\text{[EOS]}}\in\mathbb{R}^{d\times 1}$, and $w$ is the controllable
target <EOS> guidance hyperparameter. Then given a source embedding $s$, such
as a man, and a guidance prompt $g$, such as a person with mustache, we then
employ $\gamma=\sigma(s,g)$ and generate the modified image using edited
embedding $\gamma$. See Fig. 2 for an illustration of the editing operation.
This reformulation of the source embedding eliminates the necessity for
unsupervised training of new tokens or finetuning.

Figure 5: Qualitative comparison. (A) We compared our method with a prompt
concatenation baseline. We used ”a nurse” as the source prompt, and ”man,
glasses” as the target prompt for generating <EOS> guided image editing. The
prompt for the baseline is ”a nurse, man, glasses”. All images were generated
with identical initial noise. High-quality generation was facilitated by
Realistic Vision V6. (B) <EOS>-based editing can be used for content
moderation. (C) <EOS>-based editing is an effective technique for background
editing as well.

##  4 Experiments

We demonstrate the efficiency of image editing within text-to-image diffusion
models by employing the <EOS> token across diverse scenarios, such as editing
images of faces, cars, and moderating nude content. Additionally, we benchmark
these edits against state-of-the-art editing methods and conduct a user study
to quantitatively assess our technique. We used SD 1.4 for our experiments. To
generate (StableDiffusion - <EOS>) paired images at inference time, we use the
same noise under the same seed. Next, we obtain the text embeddings of source
prompt (such as ”a nurse”) and target prompt (such as ”man with eyeglasses”).
Finally, we swap the <EOS> embedding of the source prompt with the <EOS>
embedding of the target prompt. In all experiments, we used 50 steps of
denoising.

### Qualitative Comparison

We compare <EOS>-based editing with state-of-the-art image editing methods
SEGA [1], Ledits++ [2] and Cycle Diffusion [16]. For every attribute of
interest, like eyeglasses, we process it using the CLIP text encoder to
acquire the <EOS> embedding of that specific attribute. Then,
$\sigma(s,g)=[s_{\text{{<SOS>}:N}}\mid g_{\text{{<EOS>}}}]$ is employed at
inference time, where the source prompt is a headshot of a woman or a headshot
of a man (see Fig. 1 and Fig. 4). When we combine the <EOS> embedding of a
specific attribute with the source prompt, like a headshot of a woman, we are
essentially editing the underlying concept in the source prompt. Subsequently,
all images generated with the same source prompt begin to exhibit visual
elements that align with the integrated attribute. As can be seen from Fig. 4,
our method achieves comparable results with state-of-the-art methods such as
SEGA [1], Ledits++ [2] and Cycle Diffusion [16].

### Complex edits

We also test more complex edits such as altering the background of an image.
For this purpose, we take random car pictures obtained by SD-1.4 and fuse them
with <EOS> embedding of text prompt sea. In Fig. 5.C we demonstrate that while
the composition of car structures remains consistent, the successful
integration of sea information in the <EOS> embedding leads to a notable
transformation in the scene background, effectively replacing it with a
seascape. We also demonstrate the trade-off between content preservation and
edit quality in Fig.3.

Method | Edit Quality | Disentanglement
---|---|---
SEGA [1] | 2.76 | 2.64
Cycle Diff. [16] | 2.99 | 2.99
LEDITS++ [2] | 3.58 | 3.27
Ours | 3.20 | 3.12
Table 1: User Study Results. The average user responses are provided in the
table. We conduct our study within a range of 1-to-5.

### Moderating NSFW Content

We illustrate the effectiveness of <EOS> guidance during the inference stage
through its application in a NSFW (Not Safe for Work) moderation scenario (see
Fig.5.B). By incorporating <EOS> into the moderation process by
$\sigma$(<NSFW>, dressed <gender>), we aim to demonstrate its capability to
enhance content filtering and ensure a more secure and appropriate online
environment. This application not only highlights the technical prowess of
<EOS> but also emphasizes its potential impact in real-world contexts where
sensitive content moderation is crucial. As can be seen from the results, our
method can successfully moderated unsafe content while keeping the original
structure of the images preserved.

### Mean Opinion Score (MOS).

We conduct a user study with 25 participants on the Prolific
platform111https://www.prolific.com to evaluate the effectiveness of our
method in terms of edit quality and disentanglement capabilities. The
participants were shown a series of input-edit pairs and asked to evaluate
them on whether the intended edit has been applied successfully and whether
the identity of the input is preserved. For each of the questions, the
participants are asked to assign a rating within the scale of 1-to-5 where 5
corresponds to the highest score. Referring to the results demonstrated in
Table 1, our method outperforms both [1] and [2] and performs comparably with
[2].

##  5 Limitations

Biases within the CLIP can affect editing results, an issue suffered by SOTA
editing methods as well. For example, when altering a person’s eye color to
blue, even SOTA methods may inadvertently remove the beard (see the last row
of Fig.4). This behavior inherently affects the disentanglement capability of
the proposed method.

##  6 Conclusion

This paper has illuminated the previously unexplored territory of CLIP’s
capabilities as a zero-shot image editing method, leveraging its EOS token to
bridge the gap between textual prompts and visual outputs.

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Supplementary Material

##  S.1 Additional results

Please see Figure S.6

Figure S.6: We show the editing capabilities of [EOS] guidance beyond marginal
attributes. We change the the entire theme to a painting theme. All pictures
are generated with the text prompt a dog. In order to obtain the edited
versions, $\sigma$(a dog, painting) is applied.

##  S.2 Details about User Study

We ask the following questions to the users:

  * •

For edit quality: The original image is shown on the left, and the modified
image is shown on the right. How likely do you think the modified image is
depicting the same person while featuring ”Makeup” attribute? Rate from 1 (Not
at all) to 5 (Very well)

  * •

For disentanglement: The original image is shown on the left, and the modified
image is shown on the right. How likely do you think the modified image
reflects ”Makeup” feature? Rate from 1 (Not at all) to 5 (Very well)


[a]Gudrun Heinrich

# Beyond dimension six in SM Effective Field Theory:
a case study in Higgs pair production at NLO QCD

Jannis Lang  Ludovic Scyboz

###### Abstract

We present the NLO (two-loop) QCD corrections to Higgs boson pair production
in gluon fusion within Standard Model Effective Theory (SMEFT), including also
squared dimension-6 operators and double insertions of operators. The
different options to truncate the EFT expansion are contrasted to a non-linear
EFT approach (HEFT) and their effects are illustrated by several
phenomenological examples.

##  1 Introduction

Higgs-boson pair production is commonly viewed as the golden channel to
improve our understanding of the Higgs potential. The gluon fusion production
channel is the process with the largest cross section that provides direct
constraints on the Higgs trilinear self-coupling. The effects of anomalous
couplings are typically studied in the context of Effective Field Theories
(EFTs): In general, this procedure introduces a new theoretical systematic
uncertainty, associated with the truncation of the series in the EFT expansion
parameter. It is thus important to be able to reliably estimate of the size of
these uncertainties. Results for $gg\to hh$ have already been obtained in the
non-linear Higgs Effective Field Theory (HEFT) at full NLO QCD [1], and later
have been matched to parton showers in a Powheg-Box [2, 3, 4] implementation
[5]. Here, we present NLO QCD results for Higgs-boson pair production within
the Standard Model Effective Field Theory (SMEFT), based on the calculation
explained in more detail in [6]. Our formalism allows the user to choose
between different truncation options. The NLO corrections are implemented in
the public ggHH_SMEFT generator in the Powheg-Box.111Available at
https://powhegbox.mib.infn.it as User-Processes-V2/ggHH_SMEFT.

##  2 Effective field theory expansion schemes

In this section, we define our conventions and present the EFT systematics we
used. We focus on the linear Standard Model Effective field theory (SMEFT)
expansion, and contrast it to the non-linear Higgs Effective Field Theory
(HEFT, also called electroweak chiral Lagrangian).

The SMEFT [7, 8, 9] is an effective field theory with an expansion based on
counting the canonical dimension of operators, where the Wilson coefficients
of higher dimensional operators are suppressed by inverse powers of the scale
$\Lambda$, a characteristic scale of the unknown new physics. Operators are
composed of SM fields and SM symmetries are imposed, such that the Higgs field
is contained in a doublet transforming linearly under $SU(2)_{L}\times U(1)$.
In our study, we only consider the contributions of the leading dimension-6
operators and hence the Lagrangian is of the form

$\mathcal{L}_{\text{SMEFT}}=\mathcal{L}_{\text{SM}}+\sum_{i}\frac{C_{i}^{(6)}}{\Lambda^{2}}\mathcal{O}_{i}^{\rm{dim6}}+{\cal
O}(\frac{1}{\Lambda^{3}})\;.$ (1)

The number of independent operators at this level is already quite
substantial, but only a restricted subset enters in gluon fusion Higgs pair
production.

We work in the so-called Warsaw basis [8], hence the relevant dimension-6
operators are given by

$\begin{split}\Delta\mathcal{L}_{\text{Warsaw}}&=\frac{C_{H,\Box}}{\Lambda^{2}}(\phi^{\dagger}\phi)\Box(\phi^{\dagger}\phi)+\frac{C_{HD}}{\Lambda^{2}}(\phi^{\dagger}D_{\mu}\phi)^{*}(\phi^{\dagger}D^{\mu}\phi)+\frac{C_{H}}{\Lambda^{2}}(\phi^{\dagger}\phi)^{3}\\\
&+\left(\frac{C_{uH}}{\Lambda^{2}}\phi^{\dagger}{\phi}\bar{q}_{L}\phi^{c}t_{R}+h.c.\right)+\frac{C_{HG}}{\Lambda^{2}}\phi^{\dagger}\phi
G_{\mu\nu}^{a}G^{\mu\nu,a}\;.\end{split}$ (2)

The chromo-magnetic operator is omitted since it comes with an additional
loop-suppression factor relative to the operators entering eq. (2) [10, 1,
11].

In HEFT [12, 13, 14, 15, 16, 17] the chiral dimension $d_{\chi}$ [18] is used
for the classification of operators, which is formally identical to a counting
in loop orders with $d_{\chi}=2L+2$ [19, 20]. The expansion parameter is given
by $\frac{f^{2}}{\Lambda^{2}}\sim\frac{1}{16\pi^{2}}$, thus the Lagrangian can
be expressed as

$\displaystyle{\cal L}_{d_{\chi}}={\cal
L}_{(d_{\chi}=2)}+\sum_{L=1}^{\infty}\sum_{i}\left(\frac{1}{16\pi^{2}}\right)^{L}c_{i}^{(L)}O^{(L)}_{i}\;.$
(3)

The relevant terms for $gg\to hh$ up to $d_{\chi}=4$ are

$\displaystyle\Delta{\cal
L}_{\text{HEFT}}=-m_{t}\left(c_{t}\frac{h}{v}+c_{tt}\frac{h^{2}}{v^{2}}\right)\,\bar{t}\,t-c_{hhh}\frac{m_{h}^{2}}{2v}h^{3}+\frac{\alpha_{s}}{8\pi}\left(c_{ggh}\frac{h}{v}+c_{gghh}\frac{h^{2}}{v^{2}}\right)\,G^{a}_{\mu\nu}G^{a,\mu\nu}\;.$
(4)

The anomalous couplings are a priori unrelated, since the physical Higgs field
enters as a singlet under SM symmetries.

After expansion of the Higgs doublet in eq. (2) around its vacuum expectation
value and application of the field redefinition

$\displaystyle h\to
h+v^{2}\frac{C_{H,\textrm{kin}}}{\Lambda^{2}}\left(h+\frac{h^{2}}{v}+\frac{h^{3}}{3v^{2}}\right)\;,$
(5)

with

$C_{H,\textrm{kin}}:=C_{H,\Box}-\frac{1}{4}\,C_{HD}\;,$

the Higgs kinetic term acquires its canonical form (up to ${\cal
O}\left(\Lambda^{-4}\right)$ terms). Comparing terms in the Lagrangian with
eq. (4), we end up with the translation of coupling coefficients listed in
Table 1, valid at $\mathcal{O}(\Lambda^{-2})$ at the level of the Lagrangian.
However, this translation has to be considered with care, since in SMEFT the
EFT expansion is based on the assumption that $C_{i}\frac{s}{\Lambda^{2}}$ is
a small quantity, allowing only for small deviations from the SM, whereas in
HEFT the anomalous couplings $c_{i}$ can be of order ${\cal O}(1)$.

HEFT | Warsaw
---|---
$c_{hhh}$ | $1-2\frac{v^{2}}{\Lambda^{2}}\frac{v^{2}}{m_{h}^{2}}\,C_{H}+3\frac{v^{2}}{\Lambda^{2}}\,C_{H,\textrm{kin}}$
$c_{t}$ | $1+\frac{v^{2}}{\Lambda^{2}}\,C_{H,\textrm{kin}}-\frac{v^{2}}{\Lambda^{2}}\frac{v}{\sqrt{2}m_{t}}\,C_{uH}$
$c_{tt}$ | $-\frac{v^{2}}{\Lambda^{2}}\frac{3v}{2\sqrt{2}m_{t}}\,C_{uH}+\frac{v^{2}}{\Lambda^{2}}\,C_{H,\textrm{kin}}$
$c_{ggh}$ | $\frac{v^{2}}{\Lambda^{2}}\frac{8\pi}{\alpha_{s}}\,C_{HG}$
$c_{gghh}$ | $\frac{v^{2}}{\Lambda^{2}}\frac{4\pi}{\alpha_{s}}\,C_{HG}$
Table 1: Translation at Lagrangian level between operators in HEFT and SMEFT
in the Warsaw basis.

The SMEFT series is usually truncated at order $\mathcal{O}(1/\Lambda^{2})$,
and contributions from squared dimension-6 operators, as well as dimension-8
operators, are typically neglected since they are formally suppressed (of
order $\mathcal{O}(1/\Lambda^{4}$)). In order to study parts of the neglected
contributions, in the following we allow all possible dimension-6 operator
insertions at amplitude level. The amplitude can be separated into a pure SM
term (no insertion), a single insertion (dim6), and a double operator-
insertion (dim62) term, as shown in eq. (6).

$\displaystyle{\cal M}=$ $\displaystyle\
\vbox{\hbox{\includegraphics[page=1,scale={0.9}]{gghh_diagrams}}}+\vbox{\hbox{\includegraphics[page=2,scale={0.9}]{gghh_diagrams}}}+\vbox{\hbox{\includegraphics[page=3,scale={0.9}]{gghh_diagrams}}}$
$\displaystyle\
+\vbox{\hbox{\includegraphics[page=4,scale={0.9}]{gghh_diagrams}}}+\vbox{\hbox{\includegraphics[page=5,scale={0.9}]{gghh_diagrams}}}+\dots$
$\displaystyle=$ $\displaystyle\ {\cal M}_{\text{SM}}+{\cal
M}_{\rm{dim6}}+{\cal M}_{\rm{dim6}^{2}}\;.$ (6)

When taking the square of the amplitude, $\sigma\propto|\mathcal{M}|^{2}$, we
define the following four truncation options, which differ in the way the
above terms are taken into account:

$\displaystyle\sigma\simeq\left\\{\begin{aligned} &\
\sigma_{\text{SM}}+\sigma_{\text{SM}\times\rm{dim6}}&\textrm{(a)}\\\ &\
\sigma_{\left(\text{SM}+\rm{dim6}\right)\times\left(\text{SM}+\rm{dim6}\right)}&\textrm{(b)}\\\
&\
\sigma_{\left(\text{SM}+\rm{dim6}\right)\times\left(\text{SM}+\rm{dim6}\right)}+\sigma_{\text{SM}\times\rm{dim6}^{2}}&\textrm{(c)}\\\
&\
\sigma_{\left(\text{SM}+\rm{dim6}+\rm{dim6}^{2}\right)\times\left(\text{SM}+\rm{dim6}+\rm{dim6}^{2}\right)}&\textrm{(d)}\end{aligned}\right.$
(7)

Option (a) corresponds to the first order of the expansion of
$\sigma\propto|\mathcal{M}|^{2}$ in $\Lambda^{-2}$. Option (b) is the first
order of the expansion of the amplitude $\mathcal{M}$ in $\Lambda^{-2}$.
Option (c) includes all terms stemming from single- and double-insertions of
dimension-6 operators at order $\mathcal{O}(\Lambda^{-4})$, i.e. all
contributions but those from dimension-8 operators. Option (d) does not
include any linearisation whatsoever, thus corresponds to the case of HEFT
upon translation of the parameters as given in Table 1 (up to the running of
the strong coupling appearing in the effective gluon-Higgs couplings). In the
following, we investigate whether differences between these options can serve
as a proxy for estimating uncertainties related to the truncation of higher-
dimensional operators.

##  3 Cross-section results

Our results were produced for a centre-of-mass energy of $\sqrt{s}=13$ TeV,
where we used the PDF4LHC15_nlo_30_pdfas [21] parton distribution functions
interfaced to our code via LHAPDF [22], along with the corresponding value for
$\alpha_{s}$. The masses of the Higgs boson and the top quark have been fixed,
as in the virtual amplitude, to $m_{h}=125$ GeV, $m_{t}=173$ GeV and their
widths have been set to zero. Jets are clustered with the anti-$k_{T}$
algorithm [23] as implemented in the FastJet package [24, 25], with jet radius
$R=0.4$ and a minimum transverse momentum $p_{T,\mathrm{min}}^{\rm{jet}}=20$
GeV. We set the renormalisation and factorisation scales to
$\mu_{R}=\mu_{F}=m_{hh}/2$.

We show results for the SM and three benchmark points, derived originally in
Ref. [26] based on a clustering of the EFT phase space into seven
characteristic $m_{hh}$-shapes. The original benchmark points were refined to
accommodate more recent experimental constraints [27, 28], as well as the
linear SMEFT relation $c_{ggh}=2c_{gghh}$. These benchmark points, marked with
a star, are given in Table 2, with the corresponding values of the SMEFT
coefficients as obtained by the translation of Table 1 at $\Lambda=1$ TeV.

| benchmark
---
$c_{hhh}$ | $c_{t}$ | $c_{tt}$ | $c_{ggh}$ | $c_{gghh}$ | $C_{H,\textrm{kin}}$ | $C_{H}$ | $C_{uH}$ | $C_{HG}$ | $\Lambda$
SM | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |  $1\;$TeV
$1^{\ast}$ | $5.105$ | $1.1$ | $0$ | $0$ | $0$ | $4.95$ | $-6.81$ | $3.28$ | $0$ |  $1\;$TeV
$3^{\ast}$ | $2.21$ | $1.05$ | $-\frac{1}{3}$ | $0.5$ | 0.25 | $13.5$ | $2.64$ | $12.6$ | $0.0387$ |  $1\;$TeV
$6^{\ast}$ | $-0.684$ | $0.9$ | $-\frac{1}{6}$ | $0.5$ | $0.25$ | $0.561$ | $3.80$ | $2.20$ | $0.0387$ |  $1\;$TeV
Table 2: Benchmark points used for the total cross sections and the
distributions of the invariant mass of the Higgs-boson pair, cf. Table 3 and
Fig. 2. The value of $C_{HG}$ is determined using $\alpha_{s}(m_{Z})=0.118$.

Inclusive cross-sections are summarised in Table 3 for truncation option (b)
with $\Lambda=1$ TeV and $\Lambda=2$ TeV, along with results from truncation
option (a) and HEFT. As evidenced for the case of benchmark point 1, the
purely linear truncation option (a) can lead to unphysical cross sections,
which serves to conclude that this benchmark point is not a valid SMEFT point
at $\Lambda=1$ TeV.

benchmark |  $\sigma_{\rm{NLO}}$[fb] | K-factor | ratio to SM |  $\sigma_{\rm{NLO}}$[fb] |  $\sigma_{\rm{NLO}}$[fb]
---|---|---|---|---|---
point | option (b) | option (b) | option (b) | option (a) | HEFT
SM | 27.94${}^{+13.7\%}_{-12.8\%}$ | 1.67 | 1 | - | -
$\Lambda=1$ TeV
$1^{\ast}$ | 74.29${}^{+19.8\%}_{-15.6\%}$ | 2.13 | 2.66 | -61.17 | 94.32
$3^{\ast}$ | 69.20${}^{+11.7\%}_{-10.3\%}$ | 1.82 | 2.47 | 29.64 | 72.43
$6^{\ast}$ | 72.51${}^{+20.6\%}_{-16.4\%}$ | 1.90 | 2.60 | 52.89 | 91.40
$\Lambda=2$ TeV
$1^{\ast}$ | 14.03${}^{+12.0\%}_{-11.9\%}$ | 1.56 | 0.502 | 5.58 | -
$3^{\ast}$ | 30.81${}^{+16.0\%}_{-14.4\%}$ | 1.71 | 1.10 | 28.35 | -
$6^{\ast}$ | 35.39${}^{+17.5\%}_{-15.2\%}$ | 1.76 | 1.27 | 34.18 | -
Table 3: Total cross sections for Higgs-boson pair production at full NLO QCD
for three benchmark points and truncation option (b). The total cross sections
for truncation option (a) (linearised dim-6) are also given, in order to
highlight the difference, as well as the values for HEFT. The fact that
truncation option (a) leads to a negative cross section for benchmark 1
clearly indicates that this is not a valid parameter point in SMEFT for
$\Lambda=1$ TeV. The uncertainties are scale uncertainties based on 3-point
scale variations.

The ratio of the cross section to the SM value, $\sigma/\sigma_{\rm SM}$, is
shown as a function of the couplings $C_{H}$, $C_{uH}$ in Fig. 1 for the
linear (left), quadratic (middle) and HEFT-like (right) truncation options. A
large part of the parameter space is characterised by negative cross-sections
for truncation option (a) (blank patch in the left plot). The iso-contours of
the cross-section become distorted by higher-order monomials in the couplings,
when going from the linearised case to the quadratic, respectively the HEFT-
like case.

Figure 1:  Heat maps showing the dependence of the cross section on the
couplings $C_{H}$, $C_{uH}$ (left) and $C_{H}$, $C_{H,\textrm{kin}}$ (right)
with $\Lambda=1$ TeV for different truncation options. Top: option (a) (linear
dim-6), middle: option (b) (quadratic dim-6), bottom: option (d) (no
linearisation in $1/\Lambda^{2}$). The white areas denote regions in parameter
space where the corresponding cross section would be negative.

Finally, in Fig. 2 we present differential results for the Higgs-pair
invariant mass $m_{hh}$, for the benchmark points 3 (left column) and 6 (right
column) given in Table 2, at $\Lambda=1$ TeV (top row), $\Lambda=2$ TeV
(middle row) and $\Lambda=4$ TeV (bottom row). As noted previously, truncation
option (a) (dark blue) can lead to unphysical cross-sections in part of the
phase-space (here for benchmark point 3 at $\Lambda\leq 2$ TeV). We show a
3-point scale variation around the central scale $\mu_{R}=\mu_{F}=c\cdot
m_{hh}/2$, with $c\in\\{\frac{1}{2},1,2\\}$, for the SM curve (black) and
truncation option (b) (orange). We also include the HEFT curve (cyan) in the
figures for $\Lambda=1$ TeV for comparison. Truncation option (d) (dark green)
is formally equivalent to HEFT, and the difference between both curves stems
purely from the running of $\alpha_{s}$ in front of the $C_{HG}$ coefficient.
For both benchmarks, at $\Lambda=1$ TeV, truncation options (b)–(d) retain –
if only marginally – the characteristic $m_{hh}$-shape identified in Ref. [26]
(double peak separated by a dip in benchmark 3, and a shoulder left of the
peak in benchmark 6). Obviously, as the value of the heavy scale $\Lambda$ is
increased, the effect of the SMEFT operators becomes numerically suppressed,
and the differential distributions for all truncation options converge to the
SM curve.

Figure 2:  Distribution of the invariant mass $m_{hh}$ of the Higgs-boson pair
for two benchmark points of Table 2, with $\Lambda=1$ TeV (top), $\Lambda=2$
TeV (middle), and $\Lambda=4$ TeV (bottom). Left: benchmark 3∗, right:
benchmark 6∗.

##  4 Conclusions

We have presented the NLO QCD corrections to Higgs-boson pair production, with
effects of BSM physics parametrised within the SMEFT framework, including
operators up to dimension-6. The calculation is implemented in the Powheg-
Box-V2 in a flexible way, which allows different truncation options regarding
multiple insertions of dimension-6 operators as well as the possibility to
switch to the non-linear HEFT parametrisation for comparison. The results show
that a naive translation from valid HEFT anomalous coupling parameter choices
can lead to SMEFT parameters which are outside the validity range of the SMEFT
expansion, as cross sections for SMEFT truncated at linear dimension-6 level
can turn negative. Moreover, characteristic shapes of $m_{hh}$ distributions
for HEFT benchmark points can only partly be recovered by some SMEFT
truncation options after translation, since the interference patterns between
the various contributions to the amplitude are different. Points in the
coupling parameter space that are clearly within the validity range of SMEFT
tend to lead to only small distortions of the SM $m_{hh}$ distribution, such
that they are within the scale uncertainty band of the SM case, and therefore
emphasize the need for precise SM predictions.

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active stress-coupling, Figs. 7(a,b), in which cracks and Li-metal filaments
predominantly evolve across the weaker grain boundaries, the simulation with
no stress-coupling predicts cracks and Li-metal filaments to grow primarily in
a transgranular fashion. That is, in the absence of a stress-coupled reaction
driving force $\mathcal{F}$, the presence of weak grain boundaries is
virtually unseen by the electrodeposition kinetics. This difference in pattern
is expected, granted in the absence of stress-coupling, electrodeposition
kinetics is dictated by purely electrochemical forces. As discussed in Sect.
9.1, localization of higher concentration and electric potential gradients at
the tip of the straight Li-metal branch compared to the slanted grain
boundaries favors reaction and speeds up the growth of a dominant vertical Li-
metal filament across the SSE.

These results highlight the importance of coupling the electrodeposition
driving force, $\mathcal{F}$ to mechanical stress. Importantly, mechanical
stresses in the presence of microstructural heterogeneities (i.e. grain
boundaries, pores etc.) can significantly alter the morphological evolution of
cracks and Li-metal filaments across the SSE. Such coupling has also been
shown experimentally [116, 13, 12], and is reproduced here through the
proposed theoretical framework.

Having discussed the manner in which mechanics couples to electrodeposition
kinetics, we now turn our attention to the role of microstructure composition,
specifically the role of grain boundaries with varying fracture energy. We
consider a range of grain boundary fracture energies, $\psi_{\text{gb}}^{*}$,
relative to the bulk solid electrolyte fracture energy,
$\psi_{\text{bulk}}^{*}$, to explore the manner in which microstructural
features with varying mechanical properties alter the morphology of cracks and
Li-metal filaments across the SSE.

Fig. 8 shows contours of reaction coordinate (top row) and damage (bottom row)
for simulations with (a)
$\psi_{\text{gb}}^{*}=(1/5)\cdot\psi_{\text{bulk}}^{*}$, (b)
$\psi_{\text{gb}}^{*}=(1/3)\cdot\psi_{\text{bulk}}^{*}$, and
$\psi_{\text{gb}}^{*}=(1/2)\cdot\psi_{\text{bulk}}^{*}$. Note that all these
simulations are fully-coupled, and thus the result in Fig. 8(b) is identical
to that in Fig. 7(a). An animation of the simulations corresponding to Fig. 8
is shown in the Supplemental Video S4_GB_Filament_Growth.mp4.

Figure 8:  Contours of reaction coordinate $\bar{\xi}$ (top row) and damage d
(bottom row) with varying grain boundary fracture energies, respectively (a)
$\psi_{\text{gb}}^{*}=(1/5)\cdot\psi_{\text{bulk}}^{*}$, (b)
$\psi_{\text{gb}}^{*}=(1/3)\cdot\psi_{\text{bulk}}^{*}$, and (c)
$\psi_{\text{gb}}^{*}=(1/2)\cdot\psi_{\text{bulk}}^{*}$.

For both cases with $\psi_{\text{gb}}^{*}=(1/5)\cdot\psi_{\text{bulk}}^{*}$
and $\psi_{\text{gb}}^{*}=(1/3)\cdot\psi_{\text{bulk}}^{*}$, Figs. 8(a,b), we
can observe the role of weaker grain boundaries in guiding Li-metal to
preferentially deposit at these sites. Owing to their lower fracture energy,
grain boundaries are easier to damage compared to the bulk SSE. This creates a
favorable stress-state within the fractured grain boundaries, making it
favorable for growth of Li-metal filaments to follow the grain boundary
network, suppressing the growth of a vertical transgranular crack and Li-metal
branch. These results are consistent with experimental observations, which
report Li-metal filaments to preferentially grow along the grain boundaries
(i.e. integranularly) in a polycrystalline LLZO electrolyte [12, 13].

With increase in fracture energy of the grain boundaries relative to the bulk
SSE, $\psi_{\text{gb}}^{*}=(1/2)\cdot\psi_{\text{bulk}}^{*}$ in Fig. 8(c), we
observe a change in the resulting crack and Li-metal filament morphology. We
observe now the formation of a distinct vertical transgranular branch,
propagating through the SSE in advance of concurrent Li-metal growth
proceeding across the grain boundaries. As the fracture energy of the grain
boundaries increases closer to that of the electrolyte bulk, fracturing these
sites becomes more difficult, and transgranular growth becomes preferable.
This result suggests there exists a threshold fracture energy ratio, where the
electrochemical driving force overcomes the weak grain boundaries and the
morphology of Li-metal filaments growth transitions from primarily
intergranular to primarily transgranular. Clearly, the limiting case of
$\psi_{\text{gb}}^{*}=\psi_{\text{bulk}}^{*}$ is identical to the simulation
results shown in Fig. 4 and produces a single vertical Li-filament.

At present, a microstructure description of the process of Li-metal filament
growth across SSEs is missing and necessary. The theoretical framework
summarized and specialized in Sects. 6 and 7 and numerical simulations
presented here show the first framework capable of predicting the resulting
morphology of Li-metal filaments for a given SSE architecture, including a
transition from primarily intergranular to primarily transgranular filament
growth.

##  10 Concluding Remarks

We have formulated and numerically implemented a thermodynamically-consistent
electro-chemo-mechanical gradient theory, which couples electrochemical
reactions with mechanical deformation and damage in solids. The framework
models transport of ionic species across the solid host under
diffusion/migration mechanisms and concurrent electrochemical reaction at
cracks across the solid host, where ionic species are reduced to deposit a new
compound. Critically, the theory captures the interplay between growth-induced
fracture of the solid host and electrodeposition of a new material inside
cracks by tracking the damage and extent of electrodeposition using separate
phase-field variables.

While the framework is general in nature, a specialization towards modeling of
Li-metal filament growth across SSEs for application in all-solid-state
batteries is demonstrated. In particular, we elucidate the coupling of
electrical, chemical, and mechanical processes, which govern the growth of Li-
metal filaments across SSEs with and without account for microstructure
heterogeneity. Consistent with experiments, we demonstrate the ability of the
framework to numerically reproduce both transgranular and intergranular growth
of metal protrusions across the SSE microstructure.

Major contributions of this work include:

  * •

The framework enables for modeling concurrent diffusion, reaction, deformation
and damage in solids. Importantly, the theory extends the current literature
in continuum mechanics to model solids undergoing electrochemical reactions,
which are intricately coupled to damage of the underlying solid host.

The interplay between electrodeposition of new material inside damaged
(fractured) regions is modeled. Consistent with experiments, electrodeposition
is restricted within damaged regions across the solid conductor, which supply
the necessary vacant space for accommodating a new material. This feature
provides an extension to current electro-chemo-mechanical phase-field
formulations in the literature, which do not model growth-induced fracture of
SSE due to Li-deposition and indistinguishably allow for Li-metal plating
across the host.

  * •

The thermodynamically consistent treatment enables derivation of a physically-
motivated reaction driving force in which different contributions of energy,
configuration entropy, electrostatics, mechanical deformations and damage can
be readily identified. Particularly useful then is the fact that material
properties driving the electrodeposition kinetics can be calibrated from the
literature or experiments.

  * •

The framework elucidates the role of mechanical confinement on the crack-
electrodeposition morphology. Under specific mechanical boundary conditions,
we numerically reproduce the presence of cracks, which are only partially
filled with Li-metal.

  * •

Finally, we demonstrate the role of microstructural heterogeneities, in
particular grain boundaries, in dictating the morphology of cracks and Li-
metal filaments across the SSE microstructure. Both transgranular and
intergranular growth mechanisms are numerically reproduced consistent with
experiments, elucidating the critical role of mechanics on electrodeposition
kinetics and the resulting morphology of Li-metal filaments across the SSE.

Growth of Li-metal filaments across SSEs constitutes a critical degradation
mechanism, hindering commercialization of next-generation all-solid-state
batteries. It is thus critical, from both an experimental and modeling
perspective, to understand the interplay between non-uniform electrodeposition
kinetics and the SSE microstructure composition. In particular, the manner in
which size and distribution of defects and microstructural heterogeneities
ultimately govern the growth of Li-metal filaments. The proposed framework, in
conjunction with experiments, enhances our current understanding of the
electro-chemo-mechanical processes , which govern the initiation and growth of
Li-metal filaments across SSEs. From a design standpoint, this work provides
then a utilitarian numerical tool, which can help elucidate new strategies to
mitigate growth of Li-metal filaments and enable for successful
commercialization of all-solid-state batteries.

## CRediT authorship contribution statement

Donald Bistri: Design and implementation of the research, Analysis of results,
Writing of the manuscript. Claudio V. Di Leo: Design and implementation of the
research, Analysis of results, Writing of the manuscript.

## Acknowledgements

Support is acknowledged from NASA Grant Number 80NSSC21M0101.

## Appendix A - Principle of Virtual Work, Macro- and Microforce Balance Laws

We present here a detailed derivation of the macro- and microforce balance
laws for the rate-like kinematical descriptors in our theory. We consider a
list of generalized virtual velocity fields to be given by
$\mathcal{V}=(\delta\boldsymbol{\chi},\delta{\bf
F}^{\text{e}},\delta\xi,\nabla\delta\xi,\delta\text{d},\nabla\delta\text{d})$
and constrained through the kinematic relation (4.2) presented in Sect. 4.1.
For any part $\mathcal{P}$ with outward unit normal ${\bf n}_{\text{\tiny R}}$
of the reference body $\mathcal{B}$, the internal and external power are given
by

$\begin{split}\delta
W_{\text{ext}}(P,\mathcal{V})&=\int_{\partial\text{P}}{\bf t}_{\text{\tiny
R}}({\bf n}_{\text{\tiny R}})\cdot\delta\boldsymbol{\chi}\,da_{\text{\tiny
R}}+\int_{\text{P}}{\bf b}_{\text{\tiny
R}}\cdot\delta\boldsymbol{\chi}\,dv_{\text{\tiny
R}}+\int_{\partial\text{P}}\eta\delta\xi\,da_{\text{\tiny
R}}+\int_{\partial\text{P}}\gamma\delta\text{d}\,da_{\text{\tiny
R}}\\\\[4.0pt] \delta W_{\text{int}}(P,\mathcal{V})&=\int_{\text{P}}({\bf
S}^{\text{e}}\mskip-3.0mu:\mskip-2.0mu\delta{\bf F}^{\text{e}}+E\delta\xi+{\bf
G}\cdot\nabla\delta\xi+\varpi\delta\text{d}+\boldsymbol{\zeta}\cdot\nabla\delta\text{d})\,dv_{\text{\tiny
R}}\end{split}$ (A.1)

To derive the balance laws, we invoke the power balance requirement (c.f.
Sect. 4.1) and demand that $\delta W_{\text{ext}}(P,\mathcal{V})=\delta
W_{\text{int}}(P,\mathcal{V})$ for all generalized virtual velocities
$\mathcal{V}$.

First, we derive the local macroforce balance. Let
$\delta\xi=0,\nabla\delta\xi=\mathbf{0},\delta\text{d}=0,\nabla\delta\text{d}=\mathbf{0}$
such that the kinematic constraint (4.2) yields the relation $\delta{\bf
F}^{\text{e}}=(\nabla\delta\boldsymbol{\chi}){\bf F}^{-1}{\bf F}^{\text{e}}$.
For this choice, the principle of virtual power gives,

$\int_{\partial\text{P}}{\bf t}_{\text{\tiny R}}({\bf n}_{\text{\tiny
R}})\cdot\delta\boldsymbol{\chi}\,da_{\text{\tiny R}}+\int_{\text{P}}{\bf
b}_{\text{\tiny R}}\cdot\delta\boldsymbol{\chi}\,dv_{\text{\tiny
R}}=\int_{\text{P}}{\bf S}^{\text{e}}\mskip-3.0mu:\mskip-2.0mu\delta{\bf
F}^{\text{e}}\,dv_{\text{\tiny R}}=\int_{\text{P}}{\bf S}^{\text{e}}{\bf
F}^{\text{c}-\mskip-2.0mu\scriptscriptstyle\top}\mskip-3.0mu:\mskip-2.0mu\nabla\delta\boldsymbol{\chi}\,dv_{\text{\tiny
R}}$ (A.2)

which by defining

${\bf T}_{\text{\tiny R}}\overset{\text{def}}{=}{\bf S}^{\text{e}}{\bf
F}^{\text{c}-\mskip-2.0mu\scriptscriptstyle\top}$ (A.3)

and applying the divergence theorem on (A.2) leads to the macroscopic force
balance

$\text{Div}\,{\bf T}_{\text{\tiny R}}+{\bf b}_{\text{\tiny
R}}=0,\quad\text{and the traction boundary condition}\quad{\bf t}_{\text{\tiny
R}}({\bf n}_{\text{\tiny R}})={\bf T}_{\text{\tiny R}}{\bf n}_{\text{\tiny
R}}.$ (A.4)

As is standard, the Piola stress ${\bf T}_{\text{\tiny R}}$ is related to the
symmetric Cauchy stress ${\bf T}$ through

${\bf T}_{\text{\tiny R}}=J{\bf T}{\bf
F}^{-\mskip-2.0mu\scriptscriptstyle\top},$ (A.5)

and for future use we may write ${\bf S}^{\text{e}}=J{\bf T}{\bf
F}^{\text{e}-\mskip-2.0mu\scriptscriptstyle\top}$.

Subsequently, we derive the local microforce balances. The first microforce
balance associated with reaction is obtained by letting
$\delta\boldsymbol{\chi}=\mathbf{0},\delta\text{d}=0,\nabla\delta\text{d}=\mathbf{0}$,
such that (4.2) yields,

$\begin{split}{\bf S}^{\text{e}}\mskip-3.0mu:\mskip-2.0mu\delta{\bf
F}^{\text{e}}&={\bf
S}^{\text{e}}\mskip-3.0mu:\mskip-2.0mu\Big{(}-h(\bar{\xi}){\bf
F}^{\text{e}}\textbf{N}^{\text{r}}\Big{)}\delta\xi\\\ &=\Big{(}-{\bf
F}^{\text{e}\,\mskip-2.0mu\scriptscriptstyle\top}{\bf
S}^{\text{e}}\mskip-3.0mu:\mskip-2.0mu\big{(}h(\bar{\xi})\textbf{N}^{\text{r}}\big{)}\Big{)}\delta\xi\\\
&=\Big{(}-(J{\bf F}^{\text{e}\,\mskip-2.0mu\scriptscriptstyle\top}{\bf T}{\bf
F}^{\text{e}-\mskip-2.0mu\scriptscriptstyle\top})\mskip-3.0mu:\mskip-2.0mu\big{(}h(\bar{\xi})\textbf{N}^{\text{r}}\big{)}\Big{)}\delta\xi\\\
&=\Big{(}-J^{\text{c}}h(\bar{\xi}){\bf
M}^{\text{e}}\mskip-3.0mu:\mskip-2.0mu\textbf{N}^{\text{r}}\Big{)}\delta\xi\end{split}$
(A.6)

where we have defined the elastic Mandel stress as,

${\bf M}^{\text{e}}\overset{\text{def}}{=}J^{\text{e}}{\bf
F}^{\text{e}\,\mskip-2.0mu\scriptscriptstyle\top}{\bf T}{\bf
F}^{\text{e}-\mskip-2.0mu\scriptscriptstyle\top}$ (A.7)

For this choice, and accounting for (A.6), the virtual power balance then
yields

$\int_{\partial\text{P}}\eta\delta\xi da_{\text{\tiny
R}}=\int_{\text{P}}\left(-J^{\text{c}}h(\bar{\xi}){\bf
M}^{\text{e}}\mskip-3.0mu:\mskip-2.0mu\textbf{N}^{\text{r}}\delta\xi+E\delta\xi+{\bf
G}\cdot\nabla\delta\xi\right)dv_{\text{\tiny R}}.$ (A.8)

Applying the divergence theorem leads to

$\int_{\text{P}}\left(-J^{\text{c}}h(\bar{\xi}){\bf
M}^{\text{e}}\mskip-3.0mu:\mskip-2.0mu\textbf{N}^{\text{r}}\delta\xi+E\delta\xi-\text{Div}{\bf
G}\,\delta\xi\right)dv_{\text{\tiny R}}=\int_{\partial\text{P}}(\eta-{\bf
G}\cdot{\bf n}_{\text{\tiny R}})\delta\xi da_{\text{\tiny R}}$ (A.9)

Granted this relationship must hold for all $\mathcal{P}$ and all values of
$\delta\xi$, standard variational arguments yield the second microforce
balance,

$E-J^{\text{c}}h(\bar{\xi}){\bf
M}^{\text{e}}\mskip-3.0mu:\mskip-2.0mu\textbf{N}^{\text{r}}-\text{Div}{\bf
G}=0$ (A.10)

along with the corresponding boundary condition

$\eta({\bf n}_{\text{\tiny R}})={\bf G}\cdot{\bf n}_{\text{\tiny R}}.$ (A.11)

Lastly, we derive the local microforce balance associated with damage. Let
$\delta\boldsymbol{\chi}=\mathbf{0},\delta\xi=0$ and
$\nabla\delta\xi=\mathbf{0}$. In a similar fashion, the virtual power balance
statement then yields

$\int_{\partial\text{P}}\gamma\delta\text{d}\,da_{\text{\tiny
R}}=\int_{\text{P}}\left(\varpi\delta\text{d}+\boldsymbol{\zeta}\cdot\nabla\delta\text{d}\right)dv_{\text{\tiny
R}}$ (A.12)

Applying the divergence theorem on (A.12) gives,

$\int_{\text{P}}(\text{Div}\boldsymbol{\zeta}-\varpi)\delta\text{d}\,dv_{\text{\tiny
R}}+\int_{\partial\text{P}}\left(\gamma-\boldsymbol{\zeta}\cdot{\bf
n}_{\text{\tiny R}}\right)\delta\text{d}\,da_{\text{\tiny R}}=0$ (A.13)

which must hold for all $\mathcal{P}$ and all values of $\delta\text{d}$,
yielding the second microforce balance

$\text{Div}\boldsymbol{\zeta}-\varpi=0$ (A.14)

along with the corresponding boundary condition

$\gamma({\bf n}_{\text{\tiny R}})=\boldsymbol{\zeta}\cdot{\bf n}_{\text{\tiny
R}}$ (A.15)

In summary, using the principle of virtual work we have derived one macroforce
balance for the Piola stress, ${\bf T}_{\text{\tiny R}}$ as well as two
corresponding microforce balances for the stresses $\\{E,{\bf
G},\boldsymbol{\zeta},\varpi\\}$ as outlined in Sect. 4.1.

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