# Feedback-aligned Mixed LLMs for Machine Language-Molecule Translation

Dimitris Gkoumas1∗ Maria Liakata1∗

1Queen Mary University of London
<EMAIL_ADDRESS>
###### Abstract

The intersection of chemistry and Artificial Intelligence (AI) is an active
area of research focused on accelerating scientific discovery. While using
large language models (LLMs) with scientific modalities has shown potential,
there are significant challenges to address, such as improving training
efficiency and dealing with the out-of-distribution problem. Focussing on the
task of automated language-molecule translation, we are the first to use
state-of-the art (SOTA) human-centric optimisation algorithms in the cross-
modal setting, successfully aligning cross-language-molecule modals. We
empirically show that we can augment the capabilities of scientific LLMs
without the need for extensive data or large models. We conduct experiments
using only 10% of the available data to mitigate memorisation effects
associated with training large models on extensive datasets. We achieve
significant performance gains, surpassing the best benchmark model trained on
extensive in-distribution data by a large margin and reach new SOTA levels.
Additionally we are the first to propose employing non-linear fusion for
mixing cross-modal LLMs which further boosts performance gains without
increasing training costs or data needs. Finally, we introduce a fine-grained,
domain-agnostic evaluation method to assess hallucination in LLMs and promote
responsible use.

##  1 Introduction

Chemistry plays a crucial role in developing innovative scientific solutions
that are scalable and cost-effective, whether it involves pioneering new drugs
Ferguson and Gray (2018), advanced materials Kippelen and Brédas (2009), or
improving chemical processes Zhong et al. (2023). However, navigating the vast
possibilities in chemistry but also biology and materials science necessitates
the involvement of Artificial Intelligence (AI) technology to enable the
acceleration of scientific discovery AI4Science and Quantum (2023).

Multi-modal models combining language with molecules have recently gained
attention as a promising approach for studying and comprehending molecules,
offering significant potential to tackle pressing global issues Zhang et al. .
Existing work has applied successful paradigms from natural language
processing (NLP) and multimodal representation learning to the chemistry
domain. One common approach involves converting the inherent three-dimensional
structures of molecules into SMILES, which provide a mapping to symbolic
character-level representations. Subsequently, researchers have explored
learning language-molecule representations in separate yet coordinated spaces
Edwards et al. (2021, 2022); Liu et al. (2023a), in a joint space Liu et al.
(2023b), or through hybrid approaches Luo et al. (2023); Christofidellis et
al. (2023). Despite significant advancements in the field, existing work does
not effectively tackle the inherent challenges in training such models; they
often rely on low quality or noisy synthetic data and typically require
significantly more data compared to ordinary NLP tasks Edwards et al. (2021).

However training larger models and utilising extensive datasets does not
always guarantee better performance. Ongoing research in NLP is focused on
innovative training methods to enhance the performance of small-sized LLMs Liu
et al. (2024). For instance, ALMA Xu et al. (2023) fine-tuned LLaMA-2 Touvron
et al. (2023) with extensive non-English monolingual data to enhance the
model’s multilingual abilities. Subsequently, supervised fine-tuning (SFT)
with high-quality parallel data was performed to instruct the model in
generating translations, surpassing all prior moderate-sized LLMs, including
large models like GPT-3.5 Brown et al. (2020). Another successful paradigm,
Reinforcement Learning with Human Feedback (RLHF) Ouyang et al. (2022), aims
to enhance the capabilities of LLMs across multiple NLP tasks, including
machine translation. A recent trend is favouring closed-form losses that
directly operate on offline preferences Rafailov et al. (2024); Xu et al.
(2024); Ethayarajh et al. (2024), simplifying over RHLF without loss in
performance.

Here, we address challenges of effectively training robust LLMs and dealing
with the out-of-distribution problem when integrating scientific modalities,
specifically focusing on machine language-molecule translation. Our approach
is the first to experiment with human-centric optimisation objectives in the
cross-modal representation learning setting to align cross language-molecule
modals with preferred offline translations. To mitigate the memorisation
effects inherent in training on large datasets, we conduct experiments using
only 10% of the L+M-24 benchmark dataset Edwards et al. (2024), achieving new
state-of-the-art (SOTA) levels. Additionally, we propose a novel approach that
fuses cross-modal LLMs to amplify performance gains without increasing
training costs or data needs. The contributions of this work are as follows:

  * •

We obtain robust LLMs for machine language-molecule translation by
experimenting with recent shifts in RLHF within the cross-modal representation
setting, namely, direct preference optimisation (DPO), contrastive preference
optimisation (CPO), and Kahneman-Tversky optimisation (KTO) ($\S$ 3.2). To the
best of our knowledge, this is the first comprehensive study that investigates
human-centric optimisations in LLMs in the multidisciplinary field of AI for
living sciences. Our experiments show that both DPO and KTO cannot cope within
agnostic cross-modal settings. In contrast, CPO, which is invariant to biases
in these settings, achieves substantial performance gains by a large margin,
reaching new SOTA levels compared to the best benchmark, Meditron, trained on
the entire dataset ($\S$ 4.4.1).

  * •

We introduce an approach that mixes dominant-modality LLMs, based on non-
linear fusion algorithms, namely TIES and SLERP ($\S$ 3.3). This addresses
inequalities between the informativeness of different modalities as well as
inherent limitations in optimisations operating on preference data, all
without increase in training costs. We empirically demonstrate that this
approach facilitates RLHF optimisation algorithms, which, to varying degrees,
depend on how well the pre-trained model is suited to specific cross-modal
settings, thereby further boosting performance gains ($\S$ 4.4.2).

  * •

We introduce a fine-grained evaluation method with multi-aspect criteria for
assessing the presence of hallucinations in LLMs. Our domain-agnostic method
can help ensure the responsible use of chemical LLMs ($\S$ 3.4). We critically
discuss our new insights and compare them with established metrics in the
literature. We conclude by highlighting the limitations of both our evaluation
criteria and previous evaluation and discuss potential future directions for
responsibly assessing generative outputs from chemical LLMs (see Section
4.4.3).

##  2 Foundations in RLHF for LLMs

Feedback-aligned LLMs traditionally undergo fine-tuning with RLHF, wherein
human preferences serve as a reward signal in optimisation Stiennon et al.
(2020); Ouyang et al. (2022). To train a LLM with RLHF, a reinforcement
learning optimisation algorithm such as PPO Schulman et al. (2017) is
typically deployed on offline preference data, involving commonly three steps:

  * •

Roll-out: Typically, a model $\pi$ is trained for auto-regressive language
generation on a large generic corpus. This training operates under the premise
that the probability distribution of a sequence of words can be broken down
into the product of conditional distributions for the next word Radford et al.
(2019).

  * •

Evaluation: A reference model $\pi_{\text{ref}}$ is employed to optimise $\pi$
for a downstream task. Typically, the $\pi_{\text{ref}}$ model undergoes fine-
tuning with an auto-regressive objective, using data pertinent to the
downstream task. This often involves instruction tuning $\pi_{\text{ref}}$ to
regulate the generated outputs.

  * •

Optimisation: The optimisation of $\pi$ with respect to $\pi_{\text{ref}}$
operates on a triple dataset $\mathcal{D}=\\{x,y_{w},y_{l}\\}$, where $x$
represents the input, and $y_{w}$ and $y_{l}$ denote preferred and dis-
preferred outputs, respectively, such that $y_{w}\succ y_{l}$ for $x$. In
Bradley and Terry (1952), the probability of $y_{w}$ being preferred over
$y_{l}$ in pairwise comparisons can be formulated as follows:

$p^{*}(y_{w}\succ y_{l}|x)=\sigma(r^{*}(x,y_{w})-r^{*}(x,y_{l}))$ (1)

Here, $\sigma$ represents the logistic function, and $r^{*}$ denotes the
“true” reward function that underlies the preferences. As obtaining the true
reward directly from a human would be prohibitively expensive, a reward model
$r_{\phi}$ is trained to act as a surrogate. This is achieved by minimising
the negative log-likelihood in human preference data;

$\mathcal{L}(r_{\phi})=-\mathbb{E}_{(x,y_{w},y_{l})\sim\mathcal{D}}[\log\sigma(r_{\phi}(x,y_{w})-r_{\phi}(x,y_{l}))]$
(2)

Additionally, the Kullback-Leibler (KL) divergence between the outputs
generated by $\pi_{\text{ref}}$ and the parameterised $\pi_{\theta}$ models
serves as an additional reward signal, ensuring that the generated responses
closely align with the reference model. Consequently, an optimal model
$\pi_{\theta}$ is one that maximises;

$\mathbb{E}_{(x\in\mathcal{D},y\in\pi_{\theta})}[r_{\phi}(x,y)]-\beta\mathcal{D}_{\text{KL}}\bigr{(}\pi_{\theta}(y|x)||\pi_{\text{ref}}(y|x)\bigr{)}$
(3)

where $\beta$ is the temperature parameter typically $\in[0.1,0.5]$.

##### Human-aware Loss Functions (HALOs):

RLHF can present challenges due to its inherent slowness and instability,
especially in the case of highly varied outputs Zheng et al. (2024). Recently,
there has been a shift towards using closed-form losses in RLHF to align
language models (LLMs) with human preferences. These losses are predominantly
HALOs that model human biases, as discussed in Tversky and Kahneman (1992),
aiming to maximise the margin between preferred and dis-preferred generated
outputs (for the definition refer to Appx. B). This approach offers a
mathematical equivalence with RLHF, while effectively addressing its inherent
limitations.

##  3 Methodology

Existing approaches for chemical models heavily rely on expensive pre-training
using large mono-modality datasets or per-task fine-tuning for each multi-
modal task. Meanwhile, other efforts concentrate on multi-task learning,
integrating vast multi-task datasets to enhance the capabilities of chemical
models (for related work please refer to Appx. A). Here, we adopt a distinct
approach by reducing reliance on extensive data and exploring optimisation
algorithms and modality fusion techniques to enhance the capabilities of
chemical LLMs.

###  3.1 Problem Definition: Machine Language-Molecule Translation (LMolT)

Let $L$ denote the source language space and $M$ denote the target molecule
space. Consider a dataset $\mathcal{D}$, comprising pairs of source language
sequences $x$ represented in space $L$, and their corresponding target
molecule sequences $y$ in space $M$, represented as
$\mathcal{D}=\\{x^{(i)},y^{(i)}\\}_{i=1}^{N}$, where $N$ is the total number
of pairs. Here, we cast the problem as a cross-modal translation task, aiming
to learn an optimal function $f:L\leftrightarrow M$ through a model
$\pi_{\theta}$ parameterised by $\theta$. We coordinate the language and
molecule spaces through instructional modelling to regulate the translation
process across both directions, i.e., $L\rightarrow M$ and $M\rightarrow L$
(please refer to Appx. C).

###  3.2 Feedback-Aligned Optimisations Algorithms

Here we investigate approximation policies for feedback alignment optimisation
for the LMolT task.

##### Supervised Fine-Tuning (SFT):

A model is optimised by minimising the disparity between the generated and
intended target outputs. For a model parameterised by $\theta$, denoted as
$\pi_{\theta}$, the loss function is defined as the negative log-likelihood of
source-target pairs $(x,y)\sim\mathcal{D}$, and can be expressed as follows:

$\mathcal{L}_{\text{NLL}}=-\mathbb{E}_{(x,y)\sim\mathcal{D}}\bigr{[}\log\pi_{\theta}(y|x)\bigr{]}$
(4)

where the quality of $(x,y)$ is crucial to the effectiveness of the
translation model Zhao et al. (2022).

##### Direct Preference Optimisation (DPO):

DPO Rafailov et al. (2024) provides a more human-centric optimisation
objective aimed at aligning the translation model $\pi$ with human intentions.
Here $\mathcal{D}$ is an offline dataset of comparisons
$\mathcal{D}=\\{x^{(i)},y_{w}^{(i)},y_{l}^{(i)}\\}{i=1}^{N}$, where $y_{w}$
represents the preferred translation (e.g. human gold standard), and $y_{l}$
represents the dis-preferred translation (typically synthetic outputs obtained
by a translation model). The loss function for DPO is constructed as a maximum
likelihood objective for a parameterised policy $\pi_{\theta}$;

$\mathcal{L}_{\text{DPO}}(\pi_{\theta};\pi_{\text{ref}})=-\mathbb{E}_{(x,y_{w},y_{l})\sim\mathcal{D}}\Bigr{[}\log\sigma\Bigl{(}\underbrace{\beta\log\frac{\pi_{\theta}(y_{w}|x)}{\pi_{\text{ref}}(y_{w}|x)}-\beta\log\frac{\pi_{\theta}(y_{l}|x)}{\pi_{\text{ref}}(y_{l}|x)}}_{{\text{Optimal
reward }\forall(x,y)\in L\times M}}\Bigl{)}\Bigr{]}$ (5)

where $\pi_{\text{ref}}$ is the fine-tuned translation model used as the
reference, $\sigma$ is the Sigmoid function, and $\beta$ is a hyper-parameter
(§ 2).

##### Contrastive Preference Optimisation (CPO):

DPO requires high-quality data or, at the very least, appropriately fine-tuned
models. In real-world scenarios, it may not be feasible to acquire high-
quality data, especially for challenging domains such as chemistry and
biology. CPO is a general approximation of Eq.5 that addresses the limitations
of DPO by training a model to avoid generating outputs that are merely
adequate, but not perfect, tested in machine translation Xu et al. (2024).
Specifically, in CPO, Eq. 5 is effectively approximated using a uniform
reference model, which assumes equal likelihood for all possible generated
outputs;

$\mathcal{L}(\pi_{\theta};U)=-\mathbb{E}_{(x,y_{w},y_{l})\sim
D}\Bigr{[}\log\sigma\Bigl{(}\beta\log\pi_{\theta}(y_{w}|x)-\beta\log\pi_{\theta}(y_{l}|x)\Bigl{)}\Bigr{]}$
(6)

Equation 6 implies that the loss is calculated based on how well generated
outputs match this uniform distribution of possible outputs, rather than being
biased towards any particular one. To maintain $\pi_{\theta}$ close to the
preferred data distribution, a behaviour cloning (BC) Hejna et al. (2023)
regulariser is injected;

$\min_{\theta}\mathcal{L}(\pi_{\theta},U)\quad\text{s.t.}\quad\mathbb{E}_{(x,y_{w})\sim
D}\Bigr{[}\mathbb{KL}\bigl{(}\pi_{w}(y_{w}|x)||\pi_{\theta}(y_{w}|x)\bigl{)}\Bigr{]}<\epsilon,$
(7)

Here, $\epsilon$ denotes a small positive constant, and $\mathbb{KL}$
signifies the Kullback-Leibler divergence. Finally, the regulariser is
streamlined with an additional SFT term on the preferred data;

$\mathcal{L}_{\text{CPO}}=\min_{\theta}\underbrace{\mathcal{L}(\pi_{\theta},U)}_{\mathcal{L}_{\text{prefer}}}-\underbrace{\mathbb{E}_{(x,y_{w})\sim
D}\bigr{[}\log\pi_{\theta}(y_{w}|x)\bigr{]}}_{\mathcal{L}_{\text{NLL}}}$ (8)

##### Kahneman-Tversky Optimisation (KTO):

KTO Ethayarajh et al. (2024) combines the implicit reward under the RLHF
objective with the concept of loss aversion from Kahneman-Tversky prospect
theory Tversky and Kahneman (1992). Unlike DPO and CPO, KTO expects a specific
dataset format
$\mathcal{D}=\\{x^{(i)},y^{(i)},y_{\lambda}^{(i)}\\}_{i=1}^{N}$, where $x$
represents the source, $y$ denotes the output, and $y_{\lambda}$ indicates
whether $y$ is preferred or dis-preferred. Thus, KTO does not rely on
preferences but rather on knowing the desirability of output $y$. Intuitively,
KTO’s loss is determined by a value function $v\rightarrow\mathbb{R}$ applied
to a generated output $z$, with respect to a reference point $z_{\text{ref}}$
and a weighting function $w$ that assigns weights to the loss for preferred
and dis-preferred generated outputs. More specifically, for training purposes,
the value function is cast as a logistic function $\sigma$;

$v_{\text{KTO}}(x,y;\beta)=\begin{cases}\sigma\bigl{(}r_{KTO}(x,y)-z_{ref}\bigl{)}&\text{if
}y\sim y_{\lambda}=\text{preferred}\\\
\sigma\bigl{(}z_{\text{ref}}-r_{\text{KTO}}(x,y)\bigl{)}&\text{if }y\sim
y_{\lambda}=\text{dis-preferred}\end{cases}$ (9)

Here, $r_{KTO}$ is set to be the implicit reward under the RLHF objective in
Eq 5;

$r_{\text{KTO}}(x,y)=\beta\log\frac{\pi_{\theta}(y|x)}{\pi_{\text{ref}}(y|x)}$
(10)

where $\beta$ is a scale factor and $z_{\text{ref}}$ is calculated in relation
to all pairs $(x^{\prime},y^{\prime})$ as an approximation of the optimal
policy, rather than being computed individually for each pair $(x,y)$;

$z_{\text{ref}}=\mathbb{E}_{x^{\prime}\sim\mathcal{D}}\left[\beta\mathbb{KL}\bigl{(}\pi_{\theta}(y^{\prime}|x^{\prime})\|\pi_{\text{ref}}(y^{\prime}|x^{\prime})\bigl{)}\right]$
(11)

Ultimately, the KTO loss boils down to the following function:

$\mathcal{L}_{\text{KTO}}(\pi_{\theta},\pi_{\text{ref}})=\mathbb{E}_{x,y\sim\mathcal{D}}\left[w(y)(1-v_{\text{KTO}}(x,y;\beta))\right]$
(12)

where $w(y)$ represents two hyper-parameters that weigh the losses for
preferred and dis-preferred completions, respectively, as follows:

$w(y)=\begin{cases}\lambda_{P}&\text{if }y\sim y_{\lambda}=\text{preferred}\\\
\lambda_{D}&\text{if }y\sim y_{\lambda}=\text{dis-preferred}\end{cases}$ (13)

###  3.3 Mixed Cross-Modals Fusion

We propose a novel fusion approach that mixes cross-modal capabilities of LLMs
excelling in unidirectional LMolT without increasing training costs or data
needs. Our approach builds on early work in model merging, which linearly
combines weights of pre-trained models into a unified, robust model Wortsman
et al. (2022). This addresses challenges related to disparities in the
informativeness of different modalities and limitations of optimisation
algorithms requiring LLMs trained on extensive data. We have employed the
following non-linear fusion algorithms for mixing cross-modal LLMs :

  * •

Trim, Elect Sign & Merge (TIES) Yadav et al. (2024): This method addresses
inference challenges caused by parameter interference from merging different
models. It involves three steps: (1) resetting parameters that changed
minimally during fine-tuning, (2) resolving sign conflicts, and (3) merging
only the parameters aligned with the final agreed-upon sign.

  * •

Spherical Linear Interpolation (SLERP) Shoemake (1985): It is a geometric
approach that blends two models while preserving their distinctive
characteristics and curvature in high-dimensional spaces. The process consists
of three steps: (1) normalising vectors to unit length, prioritising
directions over magnitudes, (2) calculating the scale factor, which determines
the interpolation factor and the angle between vectors across the fused
models, and (3) weighting and summing the vectors to derive the interpolated
representation in the fused model.

###  3.4 Evaluation Method for Assessing Hallucination in Chemical LLMs

Previous work has employed embedding representations Jaeger et al. (2018) to
assess semantics in chemical-domain models Edwards et al. (2021);
Christofidellis et al. (2023). However, these approaches often require domain
adaptation for out-of-distribution data Edwards et al. (2024) and might yield
opaque and arbitrary outcomes Steck et al. (2024). Here, in addition to
performing evaluation according to standard generation metrics, we also
introduce fine-grained multi-aspect evaluation criteria for assessing the
presence of hallucinations111Hallucination in LLMs refers to a phenomenon
where the generated outputs are inaccurate, nonsensical, or contradictory to
the provided factual information. in generated outputs.

Specifically, for our language-to-molecule translation evaluation, we
introduce the following metrics:

  * •

$\Delta_{Len}$: Measures the deviation in character length between generated
and reference molecule pairs.

  * •

Chr-F Popović (2015): Utilises the F-score statistic for character n-gram
matches between prediction-reference pairs. Here it assesses matches in
translated molecules against their references by averaging the scores of
unigram, bigram, and trigram matches. A higher Chr-F indicates better
performance.

  * •

Win Rate: Determines the percentage of successfully generated molecules within
the test subset. Each sample in the test set is considered a win if it meets
several strict criteria: a) Chr-F score > 0.3, b) absolute length deviation
from the reference < 5, and c) a valid generated output adhering to standard
molecular structure conventions222Threshold values are determined
experimentally, while validity is automatically assessed using rdkit..

For our molecule-to-language translation evaluation, we introduce similar
metrics:

  * •

$\Delta_{Len}$: Calculates the token-length deviation in generated-reference
pairs of texts.

  * •

Natural Language Inference (NLI) Williams et al. (2018): We utilise scores
from an NLI model333We use NLI-DeBERTa based on He et al. (2021)..
Specifically, we obtain the probabilities of predictions being entailed in
references. Higher probabilities indicate better performance.

  * •

Win Rate: Measures the percentage of successfully generated outputs within the
test subset. A generated output, in regard to its corresponding reference, is
considered a win if classified by the NLI model as entailment rather than
neutral or contradiction.

##  4 Experiments

###  4.1 Data

Experiments are conducted on the L+M-24 benchmark dataset, which integrates
molecular and linguistic modalities segmented into four categories with
significant applications: biomedical; light and electricity; human interaction
and organoleptics; and agriculture and industry Edwards et al. (2024). This
dataset is created by extracting molecules and their associated chemical
properties from chemical databases. These properties are subsequently
transformed into natural language using a template composition procedure with
GPT-4.

The training and validation subsets comprise approximately 127k and 34k
language-molecule pairs, respectively. We utilise 10% of these subsets for
training and validation purposes. To implement optimisation algorithms using
offline preference data, we construct tailored triple datasets (refer to
$\S$3.2) comprising both preferred and dis-preferred outputs. The preferred
outputs are comprised of the golden references from L+M-24, while we generate
the dis-preferred outputs synthetically using MolT5 Edwards et al. (2022). For
evaluation, we randomly selected 3k unseen pairs from a distinct dataset
provided as supplementary resources for extensive training of chemical LLMs at
the Language + Molecules @ ACL2024 workshop.444Sampling is conducted from the
test subset of the separate supplementary data provided for the Language +
Molecules @ ACL2024 workshop; https://github.com/language-plus-
molecules/LPM-24-Dataset.

###  4.2 Benchmark Models

We compare our results with established language-molecule models as reported
in the literature:

  * •

TxtChem-T5 Christofidellis et al. (2023): A T5XL model trained on both
linguistic and molecule modalities with a multi-task objective across various
datasets, including the CheBI-20 dataset Edwards et al. (2022), similar to
L+M-24.

  * •

Chem-LLM Zhang et al. (2024): An InternLM2-Base-7B model, trained on an
extensive chemical domain knowledge dataset, with a direct preference
optimisation objective Rafailov et al. (2024), achieving results comparable to
GPT-4.

  * •

Meditron Chen et al. (2023): A Meditron-7B model fine-tuned on the entire
L+M-24 dataset for unidirectional language-molecule translation.

  * •

SFT-Meditron: Our supervised Meditron-7B fine-tuned on a 10% subset of L+M-24
for bi-directional machine language-molecule translation.

###  4.3 Experimental Setup

All experiments are conducted using Meditron Chen et al. (2023), a model
previously noted for its performance on L+M+24. When investigating
optimisation algorithms ($\S$ 3.2), we initialise cross-modals from Meditron
trained for unidirectional language-to-molecule translation555Initialisation
of cross-modals is based on the task reported as most challenging in Edwards
et al. (2024).. In investigations of mixed modality fusion strategies ($\S$
3.3), we obtain cross-modals from separate Meditron models trained for
unidirectional language-to-molecule and molecule-to-language translation,
fusing them in a 19:1 ratio666The experimental ratio of cross-modals aims to
maintain information integrity across them.. All models are trained with QLoRA
Dettmers et al. (2024), with experimental settings detailed in Appx. F. For
evaluation metrics please refer to Appx D

###  4.4 Results

####  4.4.1 Quantitative Analysis on Feedback-Aligned Optimisation Algorithms

Table 1 presents a summary of the molecule-to-language translation results. We
observe a notable decrease in performance for benchmark models trained on
extensive data with SFT when evaluated on unseen (out-of-distribution data).
Among the baselines, Meditron exhibited the highest performance, likely due to
its training on the entire L+M-24 dataset used in our experiments. However,
training with both DPO and KTO results in failure to translate molecules to
language when initialised from agnostic cross-modals, such as those trained
from language to molecule translation ($\S$ 4.3). This suggests that these
optimisation algorithms rely heavily on the model’s suitability for a specific
task. Our SFT demonstrated competitive performance when trained on a 10%
subset of L+M-24 with a bi-directional LMolT objective, but it underperformed
compared to Meditron, trained on the entire dataset, and lacked efficiency
(for efficiency details, refer to Appendix G). By contrast, CPO showed the
ability to cope in the agnostic cross-modal setting, achieving up to a 20%
performance increase compared to Meditron (see Table 1).

Method | $\textbf{Blue-2}\uparrow$ | $\textbf{Blue-4}\uparrow$ | $\textbf{Rouge-1}\uparrow$ | $\textbf{Rouge-2}\uparrow$ | $\textbf{Rouge-L}\uparrow$ | $\textbf{METEOR}\uparrow$
---|---|---|---|---|---|---
TxtChem-T5 | 0.08 | 0.09 | 0.19 | 0.06 | 0.17 | 0.16
Chem-LLM | 0.03 | 0.00 | 0.11 | 0.02 | 0.09 | 0.14
Meditron | 0.42 | 0.30 | 0.63 | 0.47 | 0.49 | 0.54
SFT | 0.37 | 0.26 | 0.55 | 0.40 | 0.39 | 0.61
DPO | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
CPO | 0.62 | 0.45 | 0.68 | 0.50 | 0.48 | 0.62
KTO | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
$\Delta_{CPO\text{vs}MED}$ | +20% | +19% | +5% | +3% | -1% | +8%

Table 1: Molecule-to-Language Translation Results on 3k Unseen Pairs. Arrows
next to metrics indicate the higher value the better performance. Best results
are highlighted in bold. $\Delta_{CPO\text{vs}MED}$ represents the performance
gain of our best model compared to Meditron trained on the entire data.

Performance gains in language-to-molecule translation were notably higher when
cross-modals were known (see Table 2). Both DPO and CPO demonstrated similar
performance, achieving an increased accuracy of up to 42% compared to Meditron
trained on the entire dataset. Conversely, KTO exhibited low performance even
in a known cross-modal setting. We suspect that KTO is sensitive to
overfitting (refer to Appendix G).

Method | $\textbf{BLEU}\uparrow$ | $\textbf{Levenshtein}\downarrow$ | $\textbf{MACCS FTS}\uparrow$ | $\textbf{RDK FTS}\uparrow$ | $\textbf{Morgan FTS}\uparrow$ | $\textbf{FCD}\downarrow$ | $\textbf{Validity}\uparrow$
---|---|---|---|---|---|---|---
TxtChem-T5 | 0.18 | 133.29 | 0.21 | 0.10 | 0.03 | 37.67 | 0.58
Chem-LLM | 0.04 | 732.74 | 0.00 | 0.00 | 0.00 | 59.44 | 0.19
Meditron | 0.43 | 66.16 | 0.35 | 0.29 | 0.19 | 13.64 | 0.57
SFT | 0.30 | 186.99 | 0.70 | 0.62 | 0.41 | 11.14 | 0.98
DPO | 0.72 | 42.40 | 0.77 | 0.69 | 0.49 | 10.47 | 0.99
CPO | 0.71 | 42.65 | 0.77 | 0.70 | 0.48 | 4.19 | 1.00
KTO | 0.23 | 294.63 | 0.03 | 0.03 | 0.02 | 32.64 | 0.06
$\Delta_{CPO\text{vs}MED}$ | +29% | -23.76% | +42% | +41% | +30% | -9.45% | +41%

Table 2: Language-to-Molecule Translation Results on 3k Unseen Pairs. Arrows
next to metrics indicate whether higher or lower values denote better
performance. Best results are highlighted in bold. $\Delta_{CPO\text{vs}MED}$
represents the performance gain of our best models compared to Meditron.

In summary, our experiments revealed that CPO operates independently of cross-
modal considerations, whereas DPO cannot cope with an agnostic cross-modal
setting. Contrary, KTO displayed signs of overfitting even in a known cross-
modal setting. In the following section, we explore fusion approaches aimed at
enhancing the capabilities of these algorithms within the cross-modal context.

####  4.4.2 Quantitative Analysis on Mixed Cross-Modals Fusion

Her we focus on the most effective optimisation algorithms, DPO and CPO ($\S$
4.4.1), and investigate TIES and SLERP ($\S$ 3.3) fusion algorithms in the
cross-modal context. We thus aim to address disparities between language and
molecule modalities and enhance optimisation algorithm performance. Tables 3
and 4 provide an overview of the mixed cross-modal fusion results for LMolT,
maintaining consistent training data. Combining DPO with mixed cross-modals
via TIES shows promising enhancements in molecule-to-language translation (see
$\Delta_{DPO\text{vs}TIES+DPO}$ in Table 3), yet significant performance loss
in language-to-molecule translation (see $\Delta_{DPO\text{vs}TIES+DPO}$ in
Table 4). Conversely, fusing CPO with mixed cross-modals via SLERP notably
improves molecule-to-language capabilities (see
$\Delta_{CPO\text{vs}SLERP+CPO}$ in Table 3) while minimally impacting
language-to-molecule translation performance (see
$\Delta_{CPO\text{vs}SLERP+CPO}$ in Table 4), showcasing overall gains
compared to Meditron trained on the entire dataset. Overall, we consistently
observe a trade-off in performance when fusing cross-modals, prompting future
exploration of nuanced algorithms for modelling cross language-molecule
dynamics.

Fusion | Method |  Blue-2 $\uparrow$ |  Blue-4 $\uparrow$ |  Rouge-1 $\uparrow$ |  Rouge-2 $\uparrow$ |  Rouge-L $\uparrow$ |  METEOR $\uparrow$
---|---|---|---|---|---|---|---
TIES | DPO | 0.74 | 0.53 | 0.74 | 0.54 | 0.51 | 0.70
CPO | 0.74 | 0.54 | 0.76 | 0.57 | 0.53 | 0.72
SLERP | DPO | 0.00 | 0.00 | 0.02 | 0.01 | 0.00 | 0.00
CPO | 0.73 | 0.53 | 0.76 | 0.56 | 0.53 | 0.71
$\Delta_{DPO\text{vs}TIES+DPO}$ | +74% | +53% | +74% | +54% | +51% | +70%
$\Delta_{CPO\text{vs}SLERP+CPO}$ | +11% | +8% | +8% | +6% | +5% | +9%
$\Delta_{MED\text{vs}SLERP+CPO}$ | +31% | +28% | +13% | +9% | +4% | +17%

Table 3: Mixed Cross-Modals Fusion Results in Molecule-to-Language
Translation. Best combined approaches are highlighted in bold.
$\Delta_{DPO\text{vs}TIES+DPO}$, $\Delta_{CPO\text{vs}SLERP+CPO}$, and
$\Delta_{MED\text{vs}SLERP+CPO}$ measure performance gains of the best-
combined approaches from the singular cross-modal setting of DPO, CPO, and the
benchmark Meditron, as reported in Table 1.

Fusion | Method | $\textbf{BLEU}\uparrow$ | $\textbf{Levenshtein}\downarrow$ | $\textbf{MACCS FTS}\uparrow$ | $\textbf{RDK FTS}\uparrow$ | $\textbf{Morgan FTS}\uparrow$ | $\textbf{FCD}\downarrow$ | $\textbf{Validity}\uparrow$
---|---|---|---|---|---|---|---|---
TIES | DPO | 0.32 | 93.18 | 0.31 | 0.22 | 0.19 | 19.80 | 0.42
CPO | 0.68 | 46.91 | 0.72 | 0.65 | 0.45 | 24.50 | 0.94
SLERP | DPO | 0.72 | 43.85 | 0.77 | 0.70 | 0.51 | 10.35 | 0.98
CPO | 0.71 | 44.01 | 0.73 | 0.66 | 0.45 | 11.22 | 0.95
$\Delta_{DPO\text{vs}TIES+DPO}$ | -40% | +51% | -46% | -47% | -30% | +7.33% | +58%
$\Delta_{CPO\text{vs}SLERP+CPO}$ | 0% | +1.36% | -4% | -4% | -3% | +5% | -4%
$\Delta_{MED\text{vs}SLERP+CPO}$ | +29% | -22.40% | +38% | +37% | +27% | -4.45% | +37%

Table 4: Mixed Cross-Modals Fusion Results in Language-to-Molecule
Translation. Best combined approaches are highlighted in bold.
$\Delta_{DPO\text{vs}TIES+DPO}$, $\Delta_{CPO\text{vs}SLERP+CPO}$, and
$\Delta_{MED\text{vs}SLERP+CPO}$ measure performance gains of the best-
combined approaches from the singular cross-modal setting of DPO, CPO, and the
benchmark Meditron, as reported in Table 1.

####  4.4.3 Evaluation Results on Assessing Hallucination in Chemical LLMs

Hallucinations can negatively impact the usability of LMoT models. Figure 1
illustrates the distribution of metrics introduced in $\S$ 3.4 for our best-
performing models, CPO and SLERP+CPO, in different cross-modal settings
compared to Meditron, which was trained on the entire dataset. In the cross-
language-to-molecule setting, unlike Meditron, which tends to generate
molecules shorter by an average of 27 characters, our models produced
molecules with lengths similar to the actual ones. When evaluating Char-F, we
observed a substantial rightward shift in distributions (Fig. 1 (A)),
indicating that our models are significantly better at capturing uni-, bi-,
and tri-modal character-level dynamics in molecules. In the molecule-to-
language setting, our models generated language descriptions of similar length
to the actual ones, whereas Meditron consistently produced shorter
predictions. When measuring entailment probabilities, we observed a decrease
in distributions on the left side and an increase on the right side for our
models (Fig. 1 (B)), indicating that our models understand underlying concepts
rather than merely memorising language patterns. Figure 2 (B) illustrates a
case study of this phenomenon. Additional examples can be found in Appendix E.

Figure 1: Distributions of hallucination measurements ($\S$ 3.4) across 3000
pairs for our best language-molecule translation models trained on 10% of data
against the benchmark Meditron trained on the entire one. (A) Hallucination in
language-to-molecule translation: A $\Delta_{Len}$ closer to zero and higher
Char-F scores indicate superior performance. (B) Hallucination in molecule-to-
language translation: A $\Delta_{Len}$ closer to zero and higher entailment
probabilities indicate superior performance.

We proceeded by selecting models that competed with Meditron in at least one
cross-modal setting and conducted a cross win rate comparison among them, as
illustrated in Figure 2. For language-to-molecule translation, we observed a
decline in molecule generation performance for Meditron and SFT when evaluated
across multiple criteria, as outlined in $\S$ 3.4. Conversely, CPO and
SLERP+CPO demonstrated success in the multi-aspect evaluation, although
previous SLERP+CPO showed a slight performance decrease when evaluated on
single-aspect benchmark metrics (see $\Delta_{CPO\text{vs}SLERP+CPO}$ in Table
4). For molecule-to-language translation, the NLI win rates ($\S$ 3.4) of SFT
and TIES+DPO contradicted with the performance as reported in Tables 1 and 3,
while CPO and SLERP+CPO exhibited consistency with the evaluation metrics in
those tables. Although NLI evaluation generally surpasses statistical methods
when assessing generated outputs against expected references, it still faces
limitations, including accuracy issues with lengthy texts and dependence on
the quality of training dataMcIntosh et al. (2024). In the future, we aim to
explore more nuanced methods that utilise appropriate LLMs for evaluating
generated language Min et al. (2023).

Figure 2: (A) Win rate of top-performing models in at least one cross-modal
setting against Meditron. (B) A case study demonstrating that our best models
learn to synthesise concepts rather than memorising patterns. More samples
available in Appx. E

##  5 Conclusion

We focuse on the machine language-molecule translation, aiming to empirically
tackle the challenges of effectively training robust large language models
(LLMs) and addressing the out-of-distribution problem when integrating with
scientific modalities. Our approach marks the first endeavour to experiment
with human-centric optimisation algorithms in the context of cross-modal
representation learning, aligning cross-language-molecule modalities with
preferred offline translations. To combat memorisation effects inherent in
training large models on extensive datasets, we conduct experiments utilising
only 10% of the available data. This strategy yields significant performance
enhancements, surpassing the best benchmark model trained on extensive data by
a wide margin, thus achieving new state-of-the-art levels. Additionally, we
are the first to propose employing non-linear fusion for integrating cross-
modal LLMs, addressing challenges related to disparities in the
informativeness of different modalities and limitations of optimisation
algorithms requiring LLMs trained on extensive data. This approach further
enhances performance gains without increasing training costs or data
requirements. Finally, we introduce a freely-tied domain evaluation method
with multi-aspect criteria for assessing the presence of hallucinations in
chemical LLMs. In the future, we aim to investigate more nuanced algorithms
for capturing language-molecule dynamics and evaluation methods to promote
their responsible use.

## References

  * AI4Science and Quantum [2023] Microsoft Research AI4Science and Microsoft Azure Quantum.  The impact of large language models on scientific discovery: a preliminary study using gpt-4.  _arXiv preprint arXiv:2311.07361_ , 2023.
  * Bradley and Terry [1952] Ralph Allan Bradley and Milton E Terry.  Rank analysis of incomplete block designs: I. the method of paired comparisons.  _Biometrika_ , 39(3/4):324–345, 1952.
  * Brown et al. [2020] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al.  Language models are few-shot learners.  _Advances in neural information processing systems_ , 33:1877–1901, 2020.
  * Chen et al. [2023] Zeming Chen, Alejandro Hernández Cano, Angelika Romanou, Antoine Bonnet, Kyle Matoba, Francesco Salvi, Matteo Pagliardini, Simin Fan, Andreas Köpf, Amirkeivan Mohtashami, et al.  Meditron-70b: Scaling medical pretraining for large language models.  _arXiv preprint arXiv:2311.16079_ , 2023.
  * Christofidellis et al. [2023] Dimitrios Christofidellis, Giorgio Giannone, Jannis Born, Ole Winther, Teodoro Laino, and Matteo Manica.  Unifying molecular and textual representations via multi-task language modelling.  In _International Conference on Machine Learning_ , pages 6140–6157. PMLR, 2023.
  * Dettmers et al. [2024] Tim Dettmers, Artidoro Pagnoni, Ari Holtzman, and Luke Zettlemoyer.  Qlora: Efficient finetuning of quantized llms.  _Advances in Neural Information Processing Systems_ , 36, 2024.
  * Edwards et al. [2021] Carl Edwards, ChengXiang Zhai, and Heng Ji.  Text2mol: Cross-modal molecule retrieval with natural language queries.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 595–607, 2021.
  * Edwards et al. [2022] Carl Edwards, Tuan Lai, Kevin Ros, Garrett Honke, Kyunghyun Cho, and Heng Ji.  Translation between molecules and natural language.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 375–413, 2022.
  * Edwards et al. [2024] Carl Edwards, Qingyun Wang, Lawrence Zhao, and Heng Ji.  L+ m-24: Building a dataset for language+ molecules@ acl 2024.  2024\.
  * Ethayarajh et al. [2024] Kawin Ethayarajh, Winnie Xu, Niklas Muennighoff, Dan Jurafsky, and Douwe Kiela.  Kto: Model alignment as prospect theoretic optimization.  _arXiv preprint arXiv:2402.01306_ , 2024.
  * Ferguson and Gray [2018] Fleur M Ferguson and Nathanael S Gray.  Kinase inhibitors: the road ahead.  _Nature reviews Drug discovery_ , 17(5):353–377, 2018.
  * He et al. [2021] Pengcheng He, Jianfeng Gao, and Weizhu Chen.  Debertav3: Improving deberta using electra-style pre-training with gradient-disentangled embedding sharing.  _arXiv preprint arXiv:2111.09543_ , 2021.
  * Hejna et al. [2023] Joey Hejna, Rafael Rafailov, Harshit Sikchi, Chelsea Finn, Scott Niekum, W Bradley Knox, and Dorsa Sadigh.  Contrastive prefence learning: Learning from human feedback without rl.  _arXiv preprint arXiv:2310.13639_ , 2023.
  * Jaeger et al. [2018] Sabrina Jaeger, Simone Fulle, and Samo Turk.  Mol2vec: unsupervised machine learning approach with chemical intuition.  _Journal of chemical information and modeling_ , 58(1):27–35, 2018.
  * Kippelen and Brédas [2009] Bernard Kippelen and Jean-Luc Brédas.  Organic photovoltaics.  _Energy & Environmental Science_, 2(3):251–261, 2009.
  * Liu et al. [2024] Alisa Liu, Xiaochuang Han, Yizhong Wang, Yulia Tsvetkov, Yejin Choi, and Noah A Smith.  Tuning language models by proxy.  _arXiv preprint arXiv:2401.08565_ , 2024.
  * Liu et al. [2023a] Shengchao Liu, Weili Nie, Chengpeng Wang, Jiarui Lu, Zhuoran Qiao, Ling Liu, Jian Tang, Chaowei Xiao, and Animashree Anandkumar.  Multi-modal molecule structure–text model for text-based retrieval and editing.  _Nature Machine Intelligence_ , 5(12):1447–1457, 2023a.
  * Liu et al. [2023b] Zequn Liu, Wei Zhang, Yingce Xia, Lijun Wu, Shufang Xie, Tao Qin, Ming Zhang, and Tie-Yan Liu.  Molxpt: Wrapping molecules with text for generative pre-training.  _arXiv preprint arXiv:2305.10688_ , 2023b.
  * Lu and Zhang [2022] Jieyu Lu and Yingkai Zhang.  Unified deep learning model for multitask reaction predictions with explanation.  _Journal of chemical information and modeling_ , 62(6):1376–1387, 2022.
  * Luo et al. [2023] Yizhen Luo, Kai Yang, Massimo Hong, Xingyi Liu, and Zaiqing Nie.  Molfm: A multimodal molecular foundation model.  _arXiv preprint arXiv:2307.09484_ , 2023.
  * McIntosh et al. [2024] Timothy R McIntosh, Teo Susnjak, Tong Liu, Paul Watters, and Malka N Halgamuge.  Inadequacies of large language model benchmarks in the era of generative artificial intelligence.  _arXiv preprint arXiv:2402.09880_ , 2024.
  * Min et al. [2023] Sewon Min, Kalpesh Krishna, Xinxi Lyu, Mike Lewis, Wen-tau Yih, Pang Koh, Mohit Iyyer, Luke Zettlemoyer, and Hannaneh Hajishirzi.  Factscore: Fine-grained atomic evaluation of factual precision in long form text generation.  In _Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing_ , pages 12076–12100, 2023.
  * Ouyang et al. [2022] Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, et al.  Training language models to follow instructions with human feedback.  _Advances in neural information processing systems_ , 35:27730–27744, 2022.
  * Popović [2015] Maja Popović.  chrf: character n-gram f-score for automatic mt evaluation.  In _Proceedings of the tenth workshop on statistical machine translation_ , pages 392–395, 2015.
  * Radford et al. [2019] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al.  Language models are unsupervised multitask learners.  2019\.
  * Rafailov et al. [2024] Rafael Rafailov, Archit Sharma, Eric Mitchell, Christopher D Manning, Stefano Ermon, and Chelsea Finn.  Direct preference optimization: Your language model is secretly a reward model.  _Advances in Neural Information Processing Systems_ , 36, 2024.
  * Schulman et al. [2017] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov.  Proximal policy optimization algorithms.  _arXiv preprint arXiv:1707.06347_ , 2017.
  * Schwaller et al. [2019] Philippe Schwaller, Teodoro Laino, Théophile Gaudin, Peter Bolgar, Christopher A Hunter, Costas Bekas, and Alpha A Lee.  Molecular transformer: a model for uncertainty-calibrated chemical reaction prediction.  _ACS central science_ , 5(9):1572–1583, 2019.
  * Sets [2022] Molecular Sets.  A benchmarking platform for molecular generation models., 2022.
  * Shoemake [1985] Ken Shoemake.  Animating rotation with quaternion curves.  In _Proceedings of the 12th annual conference on Computer graphics and interactive techniques_ , pages 245–254, 1985.
  * Steck et al. [2024] Harald Steck, Chaitanya Ekanadham, and Nathan Kallus.  Is cosine-similarity of embeddings really about similarity?  _arXiv preprint arXiv:2403.05440_ , 2024.
  * Stiennon et al. [2020] Nisan Stiennon, Long Ouyang, Jeffrey Wu, Daniel Ziegler, Ryan Lowe, Chelsea Voss, Alec Radford, Dario Amodei, and Paul F Christiano.  Learning to summarize with human feedback.  _Advances in Neural Information Processing Systems_ , 33:3008–3021, 2020.
  * Touvron et al. [2023] Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al.  Llama 2: Open foundation and fine-tuned chat models.  _arXiv preprint arXiv:2307.09288_ , 2023.
  * Tversky and Kahneman [1992] Amos Tversky and Daniel Kahneman.  Advances in prospect theory: Cumulative representation of uncertainty.  _Journal of Risk and uncertainty_ , 5:297–323, 1992.
  * Vaucher et al. [2020] Alain C Vaucher, Federico Zipoli, Joppe Geluykens, Vishnu H Nair, Philippe Schwaller, and Teodoro Laino.  Automated extraction of chemical synthesis actions from experimental procedures.  _Nature communications_ , 11(1):3601, 2020.
  * Williams et al. [2018] Adina Williams, Nikita Nangia, and Samuel Bowman.  A broad-coverage challenge corpus for sentence understanding through inference.  In _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers)_ , pages 1112–1122, 2018.
  * Wortsman et al. [2022] Mitchell Wortsman, Gabriel Ilharco, Samir Ya Gadre, Rebecca Roelofs, Raphael Gontijo-Lopes, Ari S Morcos, Hongseok Namkoong, Ali Farhadi, Yair Carmon, Simon Kornblith, et al.  Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time.  In _International conference on machine learning_ , pages 23965–23998. PMLR, 2022.
  * Xu et al. [2023] Haoran Xu, Young Jin Kim, Amr Sharaf, and Hany Hassan Awadalla.  A paradigm shift in machine translation: Boosting translation performance of large language models.  _arXiv preprint arXiv:2309.11674_ , 2023.
  * Xu et al. [2024] Haoran Xu, Amr Sharaf, Yunmo Chen, Weiting Tan, Lingfeng Shen, Benjamin Van Durme, Kenton Murray, and Young Jin Kim.  Contrastive preference optimization: Pushing the boundaries of llm performance in machine translation.  _arXiv preprint arXiv:2401.08417_ , 2024.
  * Yadav et al. [2024] Prateek Yadav, Derek Tam, Leshem Choshen, Colin A Raffel, and Mohit Bansal.  Ties-merging: Resolving interference when merging models.  _Advances in Neural Information Processing Systems_ , 36, 2024.
  * Zhang et al. [2024] Di Zhang, Wei Liu, Qian Tan, Jingdan Chen, Hang Yan, Yuliang Yan, Jiatong Li, Weiran Huang, Xiangyu Yue, Dongzhan Zhou, et al.  Chemllm: A chemical large language model.  _arXiv preprint arXiv:2402.06852_ , 2024.
  * [42] X Zhang, L Wang, J Helwig, Y Luo, C Fu, Y Xie, M Liu, Y Lin, Z Xu, K Yan, et al.  Artificial intelligence for science in quantum, atomistic, and continuum systems. arxiv 2023.  _arXiv preprint arXiv:2307.08423_.
  * Zhao et al. [2022] Yao Zhao, Mikhail Khalman, Rishabh Joshi, Shashi Narayan, Mohammad Saleh, and Peter J Liu.  Calibrating sequence likelihood improves conditional language generation.  In _The Eleventh International Conference on Learning Representations_ , 2022.
  * Zheng et al. [2024] Chen Zheng, Ke Sun, Hang Wu, Chenguang Xi, and Xun Zhou.  Balancing enhancement, harmlessness, and general capabilities: Enhancing conversational llms with direct rlhf.  _arXiv preprint arXiv:2403.02513_ , 2024.
  * Zhong et al. [2023] Ming Zhong, Siru Ouyang, Yizhu Jiao, Priyanka Kargupta, Leo Luo, Yanzhen Shen, Bobby Zhou, Xianrui Zhong, Xuan Liu, Hongxiang Li, et al.  Reaction miner: An integrated system for chemical reaction extraction from textual data.  In _Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 389–402, 2023.

##  Appendix A Related Work

Specialised models tailored for chemistry have emerged, including Molecular
Transformers Schwaller et al. [2019] and the RXN family Vaucher et al. [2020],
which tackle tasks such as forward reaction prediction and molecular
retrosynthesis. Yet, these models necessitate distinct architectures for
individual tasks, resulting in heightened computational demands and a
prerequisite for domain-specific proficiency in each task.

T5Chem Lu and Zhang [2022] introduces a cohesive multi-tasking framework for
chemistry, leveraging a singular model for tasks spanning reaction prediction,
regression, and classification. Nonetheless, T5Chem’s reliance on task-
specific heads confines its utility to the chemical realm, thus constraining
its adaptability across diverse sub-domains.

MolT5 Edwards et al. [2022] confronts the intricate challenge of cross-domain
generation by bridging natural language and chemistry, engaging in tasks like
text-conditional de novo molecule generation and molecule captioning.
Nonetheless, MolT5’s effectiveness is curtailed by its dependence on resource-
intensive pre-training on extensive mono-modality datasets, alongside per-task
fine-tuning for each multi-modal task. Consequently, this approach restricts
MolT5’s potential to fully exploit the multi-tasking capabilities of T5 and to
facilitate seamless information sharing between tasks.

A multi-task T5 model underwent training with a multi-task objective tailored
to handle tasks across various domains, including chemistry-based tasks
(mol2mol), textual-based tasks (text2text), and cross-domain tasks (mol2text
and text2mol) Christofidellis et al. [2023]. However, the majority of these
approaches focus on optimising specific tasks, with all of them heavily
relying on supervised fine-tuning using extensive datasets.

##  Appendix B Human-aware Loss Functions (HALOs)

######  Definition 1 (HALOs)

Let $x\in X$ and $y\in Y$ denote an input and output respectively. An
$f:(x,y)\rightarrow\mathbb{R}$ is considered a human-aware loss function if it
satisfies

$f(x,y;\theta)=t\Bigr{(}v_{f}\bigr{(}r_{\theta}(x,y)-\mathbb{E}_{x^{\prime}\sim
Q^{\prime},y^{\prime}\sim
Q^{\prime}}[r_{\theta}(x^{\prime},y^{\prime})]\bigr{)}\Bigr{)}$ (14)

with a parameterised reward function $r_{\theta}$ such that
$\forall(x_{1},y_{1}),(x_{2},y_{2})\in X\times Y$,
$r_{\theta}(x_{1},y_{1})>r_{\theta}(x_{2},y_{2})\Leftrightarrow(x_{1},y_{1})\succ_{r_{\theta}}(x_{2},y_{2})$,
reference point distributions $Q_{x}(X^{\prime})$ and
$Q_{y}(Y^{\prime}|X^{\prime})$, a value function
$v_{f}:\mathbb{R}\rightarrow\mathbb{R}$ that is monotonic non-decreasing and
concave in $(0,\infty)$, and a negative affine function $t$.

##  Appendix C Language-molecule Translation Instructions

Below is an instruction that describes a task, paired with an input that
provides further context.
Write a response that appropriately completes the request.

### Instruction: You are a researcher. You can come up captions based on your
existing knowledge.
Captions are given against the following input. You should be as detailed as
possible.

### Input: Molecule: {source molecule}
In that molecule, could you formulate a caption about?

### Response:{target caption} Instruction for molecule to language
translation, i.e., $M\rightarrow L$ Below is an instruction that describes a
task, paired with an input that provides further context.
Write a response that appropriately completes the request.

### Instruction: You are a researcher. You can come up molecule smile strings
based on your existing knowledge.
Molecule smile strings are given against the following input. You should be as
detailed as possible.

### Input: Caption: {source caption}
In that caption, could you generate a molecule smile string?

### Response: {target molecule} Instruction for language to molecule
translation, i.e., $L\rightarrow M$

##  Appendix D Evaluation Metrics

For performance evaluation, we employ established metrics from the literature
Sets [2022], Edwards et al. [2022]. In molecule-to-language translation, we
assess using BLEU-2, BLEU-4, ROUGE-1, ROUGE-2, ROUGE-L, and METEOR metrics.
For language-to-molecule translation, evaluation metrics include BLEU,
Levenshtein distance, fingerprint metrics (MACCS, RDK, and Morgan), Fréchet
ChemNet Distance (FCD), and molecule validity metrics. Annotations in the
result tables indicate whether higher or lower values indicate superior
performance.

##  Appendix E Examples of Language-Molecule Translation

Figure 3: Examples of molecules generated by our top-performing models
compared to the benchmark model, Meditron. Figure 4: Examples of generated
language by our top-performing models compared to the benchmark model,
Meditron.

##  Appendix F Experimental Settings

    args = TrainingArguments(
        output_dir=save_path,
        overwrite_output_dir=True,
        load_best_model_at_end=True,
        num_train_epochs=3,
        per_device_train_batch_size=1
        per_device_eval_batch_size=1
        gradient_accumulation_steps=64
        gradient_checkpointing=False
        optim="adamw_torch_fused",
        learning_rate=5e-5,
        max_grad_norm=0.3,
        warmup_ratio=0.1,
        lr_scheduler_type="cosine",

    )

    (
        load_in_4bit=True,
        bnb_4bit_use_double_quant=True,
        bnb_4bit_quant_type=nf64,
          bnb_4bit_compute_dtype=torch.bfloat16
    )

    (
        lora_alpha=16,
        r = 64,
        lora_dropout=0.1,
        task_type="CAUSAL_LM",
        bias=False,
        target_modules= "all-linear"
    )

Figure 5: Training configuration

##  Appendix G Training Efficiency

Training efficiency across experimental methods


OUJ-FTC-12

Prediction of the Dark Fermion Mass using Multicritical-Point Principle

Yoshiki Matsuoka

Nature and Environment, Faculty of Liberal Arts, The Open University of Japan,
Chiba 261-8586, Japan

Email<EMAIL_ADDRESS>
###### Abstract

This paper proposes a method to determine the effective potential using the
multicritical-point principle (MPP) under the additional scalar field. The MPP
is applied to the model in which a singlet dark fermion and a singlet real
scalar field are added to the Standard Model (SM) to predict the dark fermion
mass. As a result, the dark fermion mass is predicted to be about $866$–$937$
GeV.

##  1 Introduction

Although about a decade has passed since the discovery of the Higgs boson,
many unsolved problems still exist[1, 2]. In particular, the nature of the
dark matter (DM) [3] is still the major unsolved problem, and many people
continue to challenge it using various approaches[4] such as theories[5, 6, 7,
8, 9], experiments[10, 11, 12], numerical computations[13], and machine
learnings[14, 15].

About 20 years before the discovery of the Higgs, Froggatt and Nielsen
proposed what they called the multicritical-point principle (MPP) and
predicted the Higgs boson mass as $135\pm 9$ GeV with $173\pm 5$ GeV for the
top quark mass in [16, 17, 18]. So far, various proposals of the MPP have been
applied[19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. The MPP is based on the idea
that “some of extreme values of the theory’s effective potential should be
degenerate.” For $h$ taken as the Higgs field, the MPP means that the energy
density tends to degenerate in two different vacuum states,
$V_{\mathrm{\mathrm{eff}}}(h_{\mathrm{EW}})=V_{\mathrm{eff}}(M_{\mathrm{pl}})\simeq
0,\
\frac{dV_{\mathrm{eff}}(h_{\mathrm{EW}})}{dh}=\frac{dV_{\mathrm{eff}}(M_{\mathrm{pl}})}{dh}\simeq
0$, where $V_{\mathrm{\mathrm{eff}}}$ is the effective potential,
$h_{\mathrm{EW}}=246$ GeV and $M_{\mathrm{pl}}\simeq 10^{19}$ GeV. The first
equation represents the equality of the energies of the two phases, which is
the prerequisite for the coexistence of the two phases. It is important to
note that the two values of the effective potential are given on so different
scales. In order to match the values of the effective potential at
$h_{\mathrm{EW}}=246\mathrm{GeV}$ and $M_{\mathrm{pl}}\simeq
10^{19}\mathrm{GeV}$, which are clearly on significantly different scales,
$V_{\mathrm{eff}}(M_{\mathrm{pl}})$ must be about zero.

In this paper, we considered the MPP to be valid even for BSM(Beyond the
Standard Model). So, we have attempted to predict the DM mass discussed in [3,
4] using MPP in the additinal scalar field as a simple extension of the SM,
which is relatively easy to realize the MPP. We have applied the MPP to the
effective potential in the two scalar fields including the Higgs field
assuming the MPP to predict the DM mass. The result shows that the DM mass is
WIMP(Weakly Interacting Massive Particle)-like.

We organize the paper as follows. In the next section, we review the MPP. In
Section 3, we define our model and apply our MPP conditions. In Section 4, we
compare our predictions with observations. In Section 5, the summary and the
discussion are given. In Appendix A, we list the one-loop renormalization
group equations (RGEs). In Appendix B, we calculate the annihilation cross
section.

##  2 Brief review of the MPP

We briefly explain the MPP. The MPP introduced in [16] is to impose the
following conditions on the SM:

$\displaystyle
V_{\mathrm{\mathrm{eff}}}(h_{\mathrm{EW}})=V_{\mathrm{eff}}(M_{\mathrm{pl}})\simeq
0,$ (1)
$\displaystyle\frac{dV_{\mathrm{eff}}(h_{\mathrm{EW}})}{dh}=\frac{dV_{\mathrm{eff}}(M_{\mathrm{pl}})}{dh}\simeq
0$ (2)

where $V_{\mathrm{\mathrm{eff}}}$ is the one-loop effective potential,
$h_{\mathrm{EW}}=246$ GeV and $M_{\mathrm{pl}}\simeq 10^{19}$ GeV. The reason
why (1) becomes approximately zero is that $h_{\mathrm{EW}}$ is so small
compared to $M_{\mathrm{pl}}$.

We apply the dimensional regularization to $V_{\mathrm{\mathrm{eff}}}$ and
shift the spacetime dimension from $4$ to $D$ and eliminate its poles in a way
that depends on the energy scale $\mu$ and perform Wick rotation. And we set
the renormalization conditions as follows:

$\displaystyle\left.V_{\mathrm{eff}}\right|_{h=0}$ $\displaystyle=$
$\displaystyle 0,$ (3)
$\displaystyle\left.\frac{d^{2}V_{\mathrm{eff}}}{dh^{2}}\right|_{h=0}$
$\displaystyle=$ $\displaystyle 0,$ (4)
$\displaystyle\left.\frac{d^{4}V_{\mathrm{eff}}}{dh^{4}}\right|_{h=M_{t}}$
$\displaystyle=$ $\displaystyle 6\lambda(M_{t})$ (5)

where $\lambda$ is the Higgs quartic coupling constant and $M_{t}$ is the top
quark mass. Then we use $\mathrm{\overline{MS}}$ scheme [29] to remove the
$\lim_{D\to 4}(1/(4-D))$ poles, the Euler’s constant $\gamma_{E}$, and
$\ln(4\pi)$ from the one-loop effective potential $V_{\mathrm{eff}}(h)$. In
other words, we simply subtract these three values from the one-loop effective
potential.

In the SM, the one-loop effective potential in Landau gauge111Thus, the mass
of the gauge fields takes on the overall factor of $\lim_{D\to 4}(3-(4-D))$.
is given by

$\displaystyle V_{\mathrm{eff}}\left(h(\mu),\mu(t)\right)$ $\displaystyle=$
$\displaystyle\frac{\lambda(\mu)}{4}h^{4}(\mu)+\frac{1}{64\pi^{2}}\left(3\lambda(\mu)h^{2}(\mu)\right)^{2}\left(\ln\frac{3\lambda(\mu)h^{2}(\mu)}{\mu^{2}(t)}-\frac{3}{2}\right)$
(6) $\displaystyle+$
$\displaystyle\frac{3}{64\pi^{2}}\left(\sqrt{\lambda(\mu)}h(\mu)\right)^{4}\left(\ln\frac{\lambda(\mu)h^{2}(\mu)}{\mu^{2}(t)}-\frac{3}{2}\right)$
$\displaystyle+$ $\displaystyle\frac{3\times
2}{64\pi^{2}}\left(\frac{g_{2}(\mu)h(\mu)}{2}\right)^{4}\left(\ln\frac{\left(g_{2}(\mu)h(\mu)\right)^{2}}{4\mu^{2}(t)}-\frac{5}{6}\right)$
$\displaystyle+$
$\displaystyle\frac{3}{64\pi^{2}}\left(\frac{\sqrt{g_{2}^{2}(\mu)+g_{Y}^{2}(\mu)}}{2}h(\mu)\right)^{4}\left(\ln\frac{\left(g_{2}^{2}(\mu)+g_{Y}^{2}(\mu)\right)h^{2}(\mu)}{4\mu^{2}(t)}-\frac{5}{6}\right)$
$\displaystyle-$ $\displaystyle\frac{4\times
3}{64\pi^{2}}\left(\frac{y_{t}(\mu)h(\mu)}{\sqrt{2}}\right)^{4}\left(\ln\frac{\left(y_{t}(\mu)h(\mu)\right)^{2}}{2\mu^{2}(t)}-\frac{3}{2}\right)$

where $\mu$ is the renormalization scale,
$\lambda(\mu),y_{t}(\mu),g_{Y}(\mu),g_{2}(\mu)$ are the Higgs quartic coupling
constant, the top Yukawa coupling constant, and the $U(1)_{Y}$ and $SU(2)_{L}$
gauge coupling constants, each being a function of $\mu(t)$, and

$\displaystyle\mu(t)\equiv M_{t}e^{t}$ (7)

where $M_{t}$ is the top quark mass and $t$ is a certain parameter. We
consider only the contribution of the top quark, which has the large Yukawa
coupling constant among the quarks when we consider the one-loop effective
potential. $h(\mu)$ is the renormalized running field. The renormalized
running field is given by

$\displaystyle h(\mu)=he^{\Gamma(\mu)}$ (8)

where $h$ is the bare field and $\Gamma(\mu)$ is the wave function
renormalization. We do not distinguish between the bare field $h$ and the
renormalized running field $h(\mu)$ because $\Gamma(\mu)$ is so small. In
[16], Froggatt and Nielsen simply choose $\mu(t)=h(\mu)$. And they predicted
the Higgs boson mass as $135\pm 9$ GeV using the MPP conditions (1), (2) on
the Planck scale.

##  3 The Model

We consider our model and how the MPP is applied. Now we consider the
following renormalizable Lagrangian:

$\displaystyle\mathcal{L}\equiv\mathcal{L}_{\mathrm{SM}}$ $\displaystyle+$
$\displaystyle\frac{1}{2}(\partial_{\mu}S)^{2}-\frac{\lambda_{\mathrm{DM}}}{4!}S^{4}+i\overline{\chi}\gamma^{\mu}\partial_{\mu}\chi-\frac{y_{\chi}}{\sqrt{2}}\overline{\chi}S\chi+\frac{1}{2}\kappa(H^{\dagger}H)S^{2}$
(9)

where $H,S,\chi$ are the Higgs field, the singlet real scalar, and the singlet
dark Dirac fermion[30]. $\lambda_{\mathrm{DM}}(\mu),\kappa(\mu),y_{\chi}(\mu)$
are the real scalar quatric coupling constant, the scalar interaction coupling
constant, and the dark Yukawa coupling constant and they are a function of
$\mu(t)$. $H$ and $S$ can be written as follows:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ h(\mu)\end{pmatrix},$ (10)
$\displaystyle S$ $\displaystyle=$ $\displaystyle s(\mu).$ (11)

where $h(\mu)$ and $s(\mu)$ are vacuum expectation values (VEVs). We set the
renormalization conditions as follows:

$\displaystyle\left.V_{\mathrm{eff}}\right|_{s=0,h=0}$ $\displaystyle=$
$\displaystyle 0,$ (12)
$\displaystyle\left.\frac{\partial^{2}V_{\mathrm{eff}}}{\partial
h^{2}}\right|_{s=0,h=0}$ $\displaystyle=$ $\displaystyle 0,$ (13)
$\displaystyle\left.\frac{\partial^{4}V_{\mathrm{eff}}}{\partial
h^{4}}\right|_{s=0,h=M_{t}}$ $\displaystyle=$ $\displaystyle 6\lambda(M_{t}),$
(14) $\displaystyle\left.\frac{\partial^{2}V_{\mathrm{eff}}}{\partial
s^{2}}\right|_{s=0,h=0}$ $\displaystyle=$ $\displaystyle 0,$ (15)
$\displaystyle\left.\frac{\partial^{4}V_{\mathrm{eff}}}{\partial
s^{4}}\right|_{s=M_{t},h=0}$ $\displaystyle=$
$\displaystyle\lambda_{\mathrm{DM}}(M_{t}),$ (16)
$\displaystyle\left.\frac{\partial^{4}V_{\mathrm{eff}}}{\partial s^{2}\partial
h^{2}}\right|_{s=M_{t},h=M_{t}}$ $\displaystyle=$
$\displaystyle-\kappa(M_{t}),$ (17)
$\displaystyle\left.\frac{\partial^{2}V_{\mathrm{eff}}}{\partial s\partial
h}\right|_{s=0,h=0}=\left.\frac{\partial^{4}V_{\mathrm{eff}}}{\partial
s^{3}\partial
h}\right|_{s=0,h=0}=\left.\frac{\partial^{4}V_{\mathrm{eff}}}{\partial
s\partial h^{3}}\right|_{s=0,h=0}$ $\displaystyle=$ $\displaystyle 0.$ (18)

The one-loop effective potential is as follows:

$\displaystyle V_{\mathrm{eff}}\left(s(\mu),h(\mu),\mu(t)\right)$
$\displaystyle=$
$\displaystyle\frac{\lambda(\mu)}{4}h^{4}(\mu)+\frac{\lambda_{\mathrm{DM}}(\mu)}{4!}s^{4}(\mu)-\frac{1}{4}\kappa(\mu)h^{2}(\mu)s^{2}(\mu)$
(19) $\displaystyle+$
$\displaystyle\frac{1}{64\pi^{2}}\left(3\lambda(\mu)h^{2}(\mu)-\frac{\kappa(\mu)}{2}s^{2}(\mu)\right)^{2}\left(\ln\frac{3\lambda(\mu)h^{2}(\mu)-\frac{\kappa(\mu)}{2}s^{2}(\mu)}{\mu^{2}(t)}-\frac{3}{2}\right)$
$\displaystyle+$
$\displaystyle\frac{3}{64\pi^{2}}\left(\sqrt{\lambda(\mu)}h(\mu)\right)^{4}\left(\ln\frac{\lambda(\mu)h^{2}(\mu)}{\mu^{2}(t)}-\frac{3}{2}\right)$
$\displaystyle+$ $\displaystyle\frac{3\times
2}{64\pi^{2}}\left(\frac{g_{2}(\mu)h(\mu)}{2}\right)^{4}\left(\ln\frac{\left(g_{2}(\mu)h(\mu)\right)^{2}}{4\mu^{2}(t)}-\frac{5}{6}\right)$
$\displaystyle+$
$\displaystyle\frac{3}{64\pi^{2}}\left(\frac{\sqrt{g_{2}^{2}(\mu)+g_{Y}^{2}(\mu)}}{2}h(\mu)\right)^{4}\left(\ln\frac{\left(g_{2}^{2}(\mu)+g_{Y}^{2}(\mu)\right)h^{2}(\mu)}{4\mu^{2}(t)}-\frac{5}{6}\right)$
$\displaystyle-$ $\displaystyle\frac{4\times
3}{64\pi^{2}}\left(\frac{y_{t}(\mu)h(\mu)}{\sqrt{2}}\right)^{4}\left(\ln\frac{\left(y_{t}(\mu)h(\mu)\right)^{2}}{2\mu^{2}(t)}-\frac{3}{2}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{64\pi^{2}}\left(\frac{\lambda_{\mathrm{DM}}(\mu)}{2}s^{2}(\mu)-\frac{\kappa(\mu)}{{2}}h^{2}(\mu)\right)^{2}\left(\ln\frac{\frac{\lambda_{\mathrm{DM}}(\mu)}{2}s^{2}(\mu)-\frac{\kappa(\mu)}{{2}}h^{2}(\mu)}{\mu^{2}(t)}-\frac{3}{2}\right)$
$\displaystyle-$
$\displaystyle\frac{4}{64\pi^{2}}\left(\frac{y_{\chi}(\mu)s(\mu)}{\sqrt{2}}\right)^{4}\left(\ln\frac{\left(y_{\chi}(\mu)s(\mu)\right)^{2}}{2\mu^{2}(t)}-\frac{3}{2}\right)$
$\displaystyle+$
$\displaystyle\frac{2}{64\pi^{2}}\left(\kappa(\mu)h(\mu)s(\mu)\right)^{2}\left(\ln\frac{\kappa(\mu)h(\mu)s(\mu)}{\mu^{2}(t)}-\frac{3}{2}\right)$

where $s(\mu)$ and $h(\mu)$ are the renormalized running fields (8).

In this case, the singlet real scalar field $s(\mu)$ and the Higgs field
$h(\mu)$ are mixing at tree-level. The mixing angle $\theta$ is defined as

$\displaystyle\left(\begin{array}[]{c}h_{1}(\mu)\\\
h_{2}(\mu)\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{l}s(\mu)\\\
h(\mu)\end{array}\right).$ (26)

where the off-diagonal terms appear due to the interaction term by
$\kappa(\mu)$. Diagonalized forms can be to eliminate the off-diagonal terms
in the anomalous dimensions. $\theta$ is given by

$\displaystyle\tan(2\theta)=\frac{2\kappa
h_{\mathrm{EW}}s_{\mathrm{DM}}}{M_{h_{1}}^{2}-M_{h_{2}}^{2}}$ (27)

where $h_{\mathrm{EW}}$ and $s_{\mathrm{DM}}$ are VEVs on the Electroweak
scale. $M_{h_{1}}$ and $M_{h_{2}}$ are the singlet real scalar mass and the
SM-like Higgs mass. These are given by

$\displaystyle
M_{h_{1,2}}^{2}=\frac{1}{2}(M_{s}^{2}+M_{h}^{2})\mp\frac{1}{2}\sqrt{(M_{s}^{2}-M_{h}^{2})^{2}+4\kappa^{2}h_{\mathrm{EW}}^{2}s_{\mathrm{DM}}^{2}}$
(28)

where $M_{s}=\sqrt{\frac{\lambda_{\mathrm{DM}}(M_{t})}{2!}}s_{\mathrm{DM}}$
and $M_{h}=\sqrt{2\lambda(M_{t})}h_{\mathrm{EW}}$ and we consider $\theta\ll
0.1$.

The one-loop effective potential
$V_{\mathrm{eff}}\left(h_{1}(\mu),h_{2}(\mu),\mu(t)\right)$ is as follows:

$\displaystyle V_{\mathrm{eff}}\left(h_{1}(\mu),h_{2}(\mu),\mu(t)\right)$
$\displaystyle=$
$\displaystyle\frac{\lambda(\mu)}{4}(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)^{4}+\frac{\lambda_{\mathrm{DM}}(\mu)}{4!}(h_{1}(\mu)\cos\theta+h_{2}(\mu)\sin\theta)^{4}$
(29) $\displaystyle-$
$\displaystyle\frac{1}{4}\kappa(\mu)(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)^{2}(h_{1}(\mu)\cos\theta+h_{2}(\mu)\sin\theta)^{2}+\frac{1}{64\pi^{2}}$
$\displaystyle\times$
$\displaystyle\Bigg{(}3\lambda(\mu)(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)^{2}-\frac{\kappa(\mu)}{2}(h_{1}(\mu)\cos\theta+h_{2}(\mu)\sin\theta)^{2}\Bigg{)}^{2}$
$\displaystyle\times$
$\displaystyle\Bigg{(}\ln\frac{3\lambda(\mu)(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)^{2}-\frac{\kappa(\mu)}{2}(h_{1}(\mu)\cos\theta+h_{2}(\mu)\sin\theta)^{2}}{\mu^{2}(t)}$
$\displaystyle-$
$\displaystyle\frac{3}{2}\Bigg{)}+\frac{3}{64\pi^{2}}\left(\sqrt{\lambda(\mu)}(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)\right)^{4}$
$\displaystyle\times$
$\displaystyle\left(\ln\frac{\lambda(\mu)(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)^{2}}{\mu^{2}(t)}-\frac{3}{2}\right)$
$\displaystyle+$ $\displaystyle\frac{3\times
2}{64\pi^{2}}\left(\frac{g_{2}(\mu)(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)}{2}\right)^{4}$
$\displaystyle\times$
$\displaystyle\left(\ln\frac{\left(g_{2}(\mu)(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)\right)^{2}}{4\mu^{2}(t)}-\frac{5}{6}\right)$
$\displaystyle+$
$\displaystyle\frac{3}{64\pi^{2}}\left(\frac{\sqrt{g_{2}^{2}(\mu)+g_{Y}^{2}(\mu)}}{2}(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)\right)^{4}$
$\displaystyle\times$
$\displaystyle\left(\ln\frac{\left(g_{2}^{2}(\mu)+g_{Y}^{2}(\mu)\right)(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)^{2}}{4\mu^{2}(t)}-\frac{5}{6}\right)$
$\displaystyle-$ $\displaystyle\frac{4\times
3}{64\pi^{2}}\left(\frac{y_{t}(\mu)(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)}{\sqrt{2}}\right)^{4}$
$\displaystyle\times$
$\displaystyle\left(\ln\frac{\left(y_{t}(\mu)(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)\right)^{2}}{2\mu^{2}(t)}-\frac{3}{2}\right)+\frac{1}{64\pi^{2}}$
$\displaystyle\times$
$\displaystyle\Bigg{(}\frac{\lambda_{\mathrm{DM}}(\mu)}{2}(h_{1}(\mu)\cos\theta+h_{2}(\mu)\sin\theta)^{2}-\frac{\kappa(\mu)}{2}(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)^{2}\Bigg{)}^{2}$
$\displaystyle\times$
$\displaystyle\Bigg{(}\ln\frac{\frac{\lambda_{\mathrm{DM}}(\mu)}{2}(h_{1}(\mu)\cos\theta+h_{2}(\mu)\sin\theta)^{2}-\frac{\kappa(\mu)}{{2}}(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)^{2}}{\mu^{2}(t)}$
$\displaystyle-$
$\displaystyle\frac{3}{2}\Bigg{)}-\frac{4}{64\pi^{2}}\left(\frac{y_{\chi}(\mu)(h_{1}(\mu)\cos\theta+h_{2}(\mu)\sin\theta)}{\sqrt{2}}\right)^{4}$
$\displaystyle\times$
$\displaystyle\left(\ln\frac{\left(y_{\chi}(\mu)(h_{1}(\mu)\cos\theta+h_{2}(\mu)\sin\theta)\right)^{2}}{2\mu^{2}(t)}-\frac{3}{2}\right)$
$\displaystyle+$
$\displaystyle\frac{2}{64\pi^{2}}\Bigg{(}\kappa(\mu)\left(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta\right)\left(h_{1}(\mu)\cos\theta+h_{2}(\mu)\sin\theta\right)\Bigg{)}^{2}$
$\displaystyle\times$
$\displaystyle\Bigg{(}\ln\frac{\kappa(\mu)(-h_{1}(\mu)\sin\theta+h_{2}(\mu)\cos\theta)(h_{1}(\mu)\cos\theta+h_{2}(\mu)\sin\theta)}{\mu^{2}(t)}-\frac{3}{2}\Bigg{)}.$

We consider this one-loop effective potential.

First, in order to obtain $s_{\mathrm{DM}}$, we consider the following VEV
conditions on the Electroweak scale[31] around $\mu=M_{t}$222We consider
something similar to the Coleman-Weinberg mechanism.:

$\displaystyle\left.\frac{\partial V_{\mathrm{eff}}}{\partial
h_{1}}\right|_{h_{1}=s_{\mathrm{DM}},h_{2}=h_{\mathrm{EW}}}$ $\displaystyle=$
$\displaystyle\left.\frac{\partial V_{\mathrm{eff}}}{\partial
h_{2}}\right|_{h_{1}=s_{\mathrm{DM}},h_{2}=h_{\mathrm{EW}}}=0.$ (30)

Next, we consider the MPP conditions on the Planck scale around $\mu\simeq
M_{\mathrm{pl}}$ in order to get the three parameters
$\lambda_{\mathrm{DM}}(\mu),y_{\chi}(\mu),\kappa(\mu)$ as follows:

$\displaystyle\left.V_{\mathrm{eff}}\right|_{h_{1}=s_{\mathrm{pl}},h_{2}=h_{\mathrm{pl}}}$
$\displaystyle=$ $\displaystyle 0,$ (31)
$\displaystyle\left.\frac{dV_{\mathrm{eff}}}{d\mu}\right|_{h_{1}=s_{\mathrm{pl}},h_{2}=h_{\mathrm{pl}}}=\left.\frac{\partial
V_{\mathrm{eff}}}{\partial
h_{1}}\frac{dh_{1}}{d\mu}\right|_{h_{1}=s_{\mathrm{pl}},h_{2}=h_{\mathrm{pl}}}+\left.\frac{\partial
V_{\mathrm{eff}}}{\partial
h_{2}}\frac{dh_{2}}{d\mu}\right|_{h_{1}=s_{\mathrm{pl}},h_{2}=h_{\mathrm{pl}}}+\cdots$
$\displaystyle=$ $\displaystyle 0.$ (32)

We search for parameters
$\lambda_{\mathrm{DM}}(\mu),y_{\chi}(\mu),\kappa(\mu)$ that satisfy these
conditions. For this purpose, we consider the one-loop RGEs [Appendix A].
Also, we find $s_{\mathrm{pl}}$ and $h_{\mathrm{pl}}$ such that the effective
potential is minimized around $\mu\simeq M_{\mathrm{pl}}$.

We refer to [32, 33]. For the top quark mass $M_{t}=172.69\pm 0.30\
\mathrm{GeV},$ the strong gauge coupling constant $\alpha(M_{Z})=0.1179\pm
0.0010$[33], we numerically found the parameters which approximately satisfy
the VEV and the MPP conditions as follows:

$\displaystyle 2450\ \mathrm{GeV}\lesssim$ $\displaystyle s_{\mathrm{DM}}$
$\displaystyle\lesssim 2650\ \mathrm{GeV},$ (33) $\displaystyle
0.392\lesssim\lambda_{\mathrm{DM}}(M_{t})\lesssim 0.395,\ $ $\displaystyle
y_{\chi}(M_{t})$ $\displaystyle\simeq 0.5,\ \kappa(M_{t})\simeq 0.008$ (34)

where $\frac{M_{\mathrm{pl}}}{0.276}\lesssim
s_{\mathrm{pl}}\lesssim\frac{M_{\mathrm{pl}}}{0.255}$ and
$h_{\mathrm{pl}}\simeq\frac{M_{\mathrm{pl}}}{0.23}$ around $\mu\simeq
M_{\mathrm{pl}}$ and we used $M_{t}$ and $\alpha(M_{Z})$ to determine the
values of the coupling constants in the SM. $\lambda_{\mathrm{DM}}(M_{t})$,
$s_{\mathrm{DM}}$, and $s_{\mathrm{pl}}$ inequality signs come from the
standard deviations of $M_{t}$ and $\alpha(M_{Z})$ and we consider $\theta\ll
0.1$.

Next, we check the MPP conditions on the Planck scale. We use
$\frac{\mu}{0.276}\lesssim h_{1}(\mu)\lesssim\frac{\mu}{0.255}$ and
$h_{2}(\mu)\simeq\frac{\mu}{0.23}$ since we are interested in the Planck
scale. In the case of $M_{t}=172.69\ \mathrm{GeV},\ \alpha(M_{Z})=0.1189,\
\lambda_{\mathrm{DM}}(M_{t})=0.394$, $h_{1}(\mu)=\frac{\mu}{0.27}$, and
$h_{2}(\mu)=\frac{\mu}{0.23}$, the numerical results of (31) and (32) are
shown in Figure 1. We can see that these values approximately satisfy the MPP
conditions.

Figure 1: The $x$-axis and $y$-axis represent
$\log_{10}(\frac{\mu(t)}{\mathrm{GeV}})$ and the value of each line. The blue
line and orange line are $\frac{V_{\mathrm{eff}}}{\mu^{4}}$ and
$\frac{1}{\mu^{3}}\frac{dV_{\mathrm{eff}}}{d\mu}$ in the case of
$M_{t}=172.69\ \mathrm{GeV},\ \alpha(M_{Z})=0.1189,\
\lambda_{\mathrm{DM}}(M_{t})=0.394$, $h_{1}(\mu)=\frac{\mu}{0.27}$, and
$h_{2}(\mu)=\frac{\mu}{0.23}$. The red line represents zero on the $y$-axis.

Figure 2 shows $\lambda(\mu)$ for the same parameters as Figure 1 in the case
of $M_{t}=172.69\ \mathrm{GeV}$, $\alpha(M_{Z})=0.1189,\
\lambda_{\mathrm{DM}}(M_{t})=0.394$, $h_{1}(\mu)=\frac{\mu}{0.27}$, and
$h_{2}(\mu)=\frac{\mu}{0.23}$. In other words, we check the vacuum stability,
so to speak, whether $\lambda(\mu)$ is negative up to the Planck scale.
$\lambda(\mu)$ reaches zero before it directly reaches zero on the Planck
scale. The vacuum stability remains meta-stable as in the SM.

Figure 2: The $x$-axis and $y$-axis represent
$\log_{10}(\frac{\mu(t)}{\mathrm{GeV}})$ and the value of $\lambda(\mu)$ in
the case of $M_{t}=172.69\ \mathrm{GeV},\ \alpha(M_{Z})=0.1189,\
\lambda_{\mathrm{DM}}(M_{t})=0.394$, $h_{1}(\mu)=\frac{\mu}{0.27}$, and
$h_{2}(\mu)=\frac{\mu}{0.23}$. The red line represents zero on the $y$-axis.

##  4 Calculations of each value

In this section, we calculate the masses, the decay rate, and the annihilation
cross sections. The singlet real scalar mass is

$\displaystyle 1084\ \mathrm{GeV}\lesssim M_{h_{1}}\lesssim 1177\
\mathrm{GeV}.$ (35)

The DM mass is

$\displaystyle 866\ \mathrm{GeV}\lesssim M_{\chi}\lesssim 937\ \mathrm{GeV}$
(36)

where
$M_{\chi}=\frac{y_{\chi}(M_{t})}{\sqrt{2}}(h_{1}\cos\theta+h_{2}\sin\theta)$.
This is a WIMP-like DM.333This result effectively reproduces the approximate
equation for the DM relic abundunce, $\Omega_{\mathrm{DM}}h^{2}\sim
0.1({M_{\chi}^{2}\over O(1)\mathrm{TeV}^{2}})(\frac{O(1)^{4}}{g_{2}^{4}})$.
The Features of WIMPs are that it is so stable, the interactions are
sufficiently weak, it is cold, and there is a suitable creation mechanism in
the early universe.

The singlet real scalar boson decays to the SM-like Higgs bosons:

$\displaystyle\Gamma(h_{1}\rightarrow
h_{2}h_{2})\simeq\frac{\kappa^{2}s_{\mathrm{DM}}^{2}}{32\pi
M_{h_{1}}}\left(1-\frac{4M_{h_{2}}^{2}}{M_{h_{1}}^{2}}\right)^{\frac{1}{2}}.$
(37)

The LHC searches for Higgs-like scalars. At present, the mixing angle $\theta$
is restricted to $\theta\lesssim 0.3$. In our model with the application of
the MPP conditions, the mixing angle $\theta$ is limited to $\theta\ll 0.1$.
This satisfies the observation limits.

Next, we calculate the annihilation cross section between the DM and each SM
particle because the DM can pair annihilate into a pair of the SM particles.
If this DM can pair annihilate into a pair of the SM fermions, then the
annihilation cross section is

$\displaystyle(\sigma v_{\mathrm{rel}})_{\chi\overline{\chi}\to
f\overline{f}}$ $\displaystyle\simeq$
$\displaystyle\frac{y_{\chi}^{2}y_{f}^{2}}{32\pi M_{\chi}^{2}}(\sin
2\theta)^{2}\left(M_{\chi}^{2}\left(M_{\chi}^{2}-M_{f}^{2}\right)\right)$ (38)
$\displaystyle\times$
$\displaystyle\left[\frac{1}{4M_{\chi}^{2}-M_{h_{1}}^{2}}-\frac{1}{4M_{\chi}^{2}-M_{h_{2}}^{2}}\right]^{2}\left(1-\frac{M_{f}^{2}}{M_{\chi}^{2}}\right)^{\frac{1}{2}}.$

For the cases of pair annihilations into a pair of the SM massive gauge
bosons, the annihilation cross sections are

$\displaystyle(\sigma v_{\mathrm{rel}})_{\chi\overline{\chi}\to W^{+}W^{-}}$
$\displaystyle\simeq$ $\displaystyle\frac{y_{\chi}^{2}g_{2}^{2}}{128\pi
M_{\chi}^{2}}(\sin
2\theta)^{2}\left(4M_{\chi}^{4}-4M_{W}^{2}M_{\chi}^{2}+3M_{W}^{4}\right)$ (39)
$\displaystyle\times$
$\displaystyle\left[\frac{1}{4M_{\chi}^{2}-M_{h_{1}}^{2}}-\frac{1}{4M_{\chi}^{2}-M_{h_{2}}^{2}}\right]^{2}\left(1-\frac{M_{W}^{2}}{M_{\chi}^{2}}\right)^{\frac{1}{2}},$
$\displaystyle(\sigma v_{\mathrm{rel}})_{\chi\overline{\chi}\to ZZ}$
$\displaystyle\simeq$
$\displaystyle\frac{y_{\chi}^{2}\left(g_{2}^{2}+g_{Y}^{2}\right)}{256\pi
M_{\chi}^{2}}(\sin
2\theta)^{2}\left(4M_{\chi}^{4}-4M_{Z}^{2}M_{\chi}^{2}+3M_{Z}^{4}\right)$ (40)
$\displaystyle\times$
$\displaystyle\left[\frac{1}{4M_{\chi}^{2}-M_{h_{1}}^{2}}-\frac{1}{4M_{\chi}^{2}-M_{h_{2}}^{2}}\right]^{2}\left(1-\frac{M_{Z}^{2}}{M_{\chi}^{2}}\right)^{\frac{1}{2}}.$

Also, for the cases of pair annihilations into a pair of the SM-like Higgs
bosons, the annihilation cross sections are

$\displaystyle(\sigma v_{\mathrm{rel}})_{\chi\overline{\chi}\to h_{2}h_{2}}$
$\displaystyle\simeq$
$\displaystyle\frac{y_{\chi}^{2}\kappa^{2}s_{\mathrm{DM}}^{2}\left(2M_{\chi}^{2}-M_{h_{2}}^{2}\right)^{2}\left(8M_{\chi}^{2}-2M_{h_{2}}^{2}\right)^{2}}{256\pi\left(8M_{\chi}^{4}-6M_{\chi}^{2}M_{h_{2}}^{2}+M_{h_{2}}^{4}\right)^{2}\left(4M_{\chi}^{2}-M_{h_{1}}^{2}\right)^{2}}\times\left(1-\frac{M_{h_{2}}^{2}}{M_{\chi}^{2}}\right)^{\frac{1}{2}}.$
(41)

We calculated the pair annihilation into a pair of the SM particles for each
case.

##  5 Summary and Discussion

We discussed the MPP using the additinal scalar field. We proposed the MPP
based on the model where the singlet dark fermion and the singlet real scalar
field are added to the SM. We placed four conditions in the model and
determined six parameters $s_{\mathrm{DM}},\ s_{\mathrm{pl}},\
h_{\mathrm{pl}},\ \lambda_{\mathrm{DM}},\ \kappa,\ y_{\chi}$.

In this case, for $M_{t}=172.69\pm 0.30\ \mathrm{GeV}$ and
$\alpha(M_{Z})=0.1179\pm 0.0010$, we found $2450\ \mathrm{GeV}\lesssim
s_{\mathrm{DM}}\lesssim 2650\ \mathrm{GeV},\
\frac{M_{\mathrm{pl}}}{0.276}\lesssim
s_{\mathrm{pl}}\lesssim\frac{M_{\mathrm{pl}}}{0.255},\
h_{\mathrm{pl}}\simeq\frac{M_{\mathrm{pl}}}{0.23},\
0.392\lesssim\lambda_{\mathrm{DM}}(M_{t})\lesssim 0.395,\
y_{\chi}(M_{t})\simeq 0.5,\ \kappa(M_{t})\simeq 0.008$. The DM mass is $866\
\mathrm{GeV}\lesssim M_{\chi}\lesssim 937\ \mathrm{GeV}$.

However, $\lambda$ does not directly reach zero on the Planck scale, but it
becomes negative just before the Planck scale. This does not solve the vacuum
stability problem and it is the problem to be considered in the future.

Also, in the extended model of the SM that we considered in this study, we
found that the DM becomes a WIMP-like DM by applying the MPP. These parameters
successfully circumvent the current restrictions, affirm the WIMP, and are in
good agreement with the WIMP’s mass range. WIMPs cover the mass range around
the Electroweak scale, which is extremely wide. Imposing the MPP conditions,
it is possible to restrict the mass range to an extremely narrow range.

Finally, we briefly discuss the future prospects of our research. Our
extension is a simple extension, and other extensions may also be able to
realize the MPP. In short, if our model is excluded by experiments, it does
not mean that the MPP has been eliminated in BSM. For example, we can naively
introduce gauge symmetry into the dark sector. In addition, although the
scalar field $S$ in our model has $Z_{2}$ symmetry, it would be so interesting
to realize the MPP in a model that removes such symmetry. Realizing the MPP in
various models is so important in searches for the DM. Additionally, it will
be important to understand the theoretical meaning of the MPP.

## Acknowledgments

We would like to thank Noriaki Aibara, So Katagiri, Akio Sugamoto, and Ken
Yokoyama for helpful comments. Especially, we are indebted to So Katagiri,
Shiro Komata, and Akio Sugamoto for reading this paper and giving useful
comments.

##  Appendix Appendix A One-loop Renormalization Group Equations

The one-loop RGEs in our Lagrangian (9) are as follows:

$\displaystyle\frac{dg_{Y}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{41}{6}g_{Y}^{3},$ (42)
$\displaystyle\frac{dg_{2}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{-19}{6}g_{2}^{3},$ (43)
$\displaystyle\frac{dg_{3}}{dt}$ $\displaystyle=$
$\displaystyle\frac{-7}{16\pi^{2}}g_{3}^{3},$ (44)
$\displaystyle\frac{dy_{t}}{dt}$ $\displaystyle=$
$\displaystyle\frac{y_{t}}{16\pi^{2}}\left(\frac{9}{2}y_{t}^{2}-\frac{17}{12}g_{Y}^{2}-\frac{9}{4}g_{2}^{2}-8g_{3}^{2}\right),$
(45) $\displaystyle\frac{dy_{\chi}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{3}{2}y_{\chi}^{3},$ (46)
$\displaystyle\frac{d\lambda}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left(\lambda\left(24\lambda-3g_{Y}^{2}-9g_{2}^{2}+12y_{t}^{2}\right)+\frac{\kappa^{2}}{2}+\frac{3}{8}g_{Y}^{4}+\frac{3}{4}g_{Y}^{2}g_{2}^{2}+\frac{9}{8}g^{4}_{2}-6y_{t}^{4}\right),$
(47) $\displaystyle\frac{d\lambda_{\mathrm{DM}}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left(\lambda_{\mathrm{DM}}\left(3\lambda_{\mathrm{DM}}+4y_{\chi}^{2}\right)+3\kappa^{2}-12y_{\chi}^{4}\right),$
(48) $\displaystyle\frac{d\kappa}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left(\kappa\left(-4\kappa+6\lambda+\lambda_{\mathrm{DM}}-\frac{3}{2}g_{Y}^{2}-\frac{9}{2}g_{2}^{2}+6y_{t}^{2}+2y_{\chi}^{2}\right)\right)$
(49)

where $t$ is a certain parameter.

##  Appendix Appendix B Calculation of the annihilation cross section

We calculate equation (38) as an example of the formula for the annihilation
cross section of the DM. In the case of non-relativistic DM, the annihilation
cross section for $\chi\overline{\chi}\to f\overline{f}$ is given by

$\displaystyle(\sigma v_{\mathrm{rel}})_{\chi\overline{\chi}\to
f\overline{f}}$ $\displaystyle=$ $\displaystyle\frac{1}{32\pi
M_{\chi}^{2}}\left(1-\frac{M_{f}^{2}}{M_{\chi}^{2}}\right)^{\frac{1}{2}}\overline{|\mathcal{M}|^{2}}$
(50)

with

$\displaystyle\overline{|\mathcal{M}|^{2}}_{\chi\overline{\chi}\to
f\overline{f}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\times\frac{y_{\chi}^{2}y_{f}^{2}}{4}\left|\sum_{i=1,2}\theta_{f,i}\frac{\overline{v}_{\chi}(p_{2})\theta_{\chi,i}u_{\chi}(p_{1})}{\left(p_{1}+p_{2}\right)^{2}-M_{h_{i}}^{2}}\right|^{2}\left|\overline{u}_{f}(k_{1})v_{f}(k_{2})\right|^{2}$
(51) $\displaystyle\simeq$ $\displaystyle\frac{y_{\chi}^{2}y_{f}^{2}}{4}(\sin
2\theta)^{2}\left(2\left(M_{\chi}^{2}-M_{f}^{2}\right)+\frac{1}{2}M_{\chi}^{2}v_{\mathrm{rel}}^{2}\right)$
$\displaystyle\times$
$\displaystyle\left(2M_{\chi}^{2}+\frac{1}{2}M_{\chi}^{2}v_{\mathrm{rel}}^{2}\right)\times\left[\frac{1}{4M_{\chi}^{2}-M_{h_{1}}^{2}}-\frac{1}{4M_{\chi}^{2}-M_{h_{2}}^{2}}\right]^{2}$

where $\theta_{\chi,1}=\theta_{f,2}=\cos\theta$ and
$\theta_{\chi,2}=-\theta_{f,1}=\sin\theta$. The velocity of the DM
$v_{\mathrm{rel}}$ is small.

## References

  * [1] G. Aad et al. [ATLAS Collaboration], “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716, 1 (2012), arXiv:1207.7214 [hep-ex].
  * [2] S. Chatrchyan et al. [CMS Collaboration], “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,” Phys. Lett. B 716, 30 (2012), arXiv:1207.7235 [hep-ex].
  * [3] L. Roszkowski, E. M. Sessolo, and S. Trojanowski, “WIMP dark matter candidates and searches— current status and future prospects,” Rept. Prog. Phys. 81 no. 6 066201 (2018), arXiv:1707.06277 [hep-ph].
  * [4] G. Arcadi, D. Cabo-Almeida, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, J. P. Neto, M. Pierre, S. Profumo, and F. S. Queiroz, “The Waning of the WIMP: Endgame?,” arXiv:2403.15860 [hep-ph].
  * [5] J. McDonald, “Gauge Singlet Scalars as Cold Dark Matter,” Phys. Rev. D 50, 3637 (1994), [hep-ph/0702143].
  * [6] S. Baek, P. Ko, W.-I. Park, et al., “Higgs Portal Vector Dark Matter : Revisited,” JHEP 05, 036 (2013), arXiv:1212.2131 [hep-ph].
  * [7] M. R. Buckley, D. Feld, and D. Goncalves, “Scalar Simplified Models for Dark Matter,” Phys. Rev. D 91, 015017 (2015), arXiv:1410.6497 [hep-ph].
  * [8] J. Abdallah, et al., “Simplified Models for Dark Matter Searches at the LHC,” Phys. Dark Univ. 9-10, 8 (2015), arXiv:1506.03116 [hep-ph].
  * [9] Y.-J. Kang, H. M. Lee, A. G. Menkara, and J. Song, “Flux-mediated dark matter,” J. High Energy Phys. 06, 013 (2021), arXiv:2103.07592 [hep-ph].
  * [10] G. Aad et al. [ATLAS Collaboration], “Combined measurements of Higgs boson production and decay using up to $80\mathrm{fb}^{-1}$ of proton-proton collision data at $\sqrt{s}=13\mathrm{TeV}$ collected with the ATLAS experiment,” Phys. Rev. D 101, 012002 (2020), arXiv:1909.02845 [hep-ex].
  * [11] [CMS Collaboration], “Combined Higgs boson production and decay measurements with up to $137\mathrm{fb}^{-1}$ of proton-proton collision data at $\sqrt{s}=13\mathrm{TeV}$,” CMS-PAS-HIG-19-005 (2020). https://cds.cern.ch/record/2706103
  * [12] [ATLAS Collaboration], “Combination of searches for invisible decays of the Higgs boson using $139\mathrm{fb}^{-1}$ of proton-proton collision data at $\sqrt{s}=13\mathrm{TeV}$ collected with the ATLAS experiment,” Phys. Lett. B 842, 137963 (2023), arXiv:2301.10731 [hep-ex].
  * [13] M. Hindmarsh and O. Philipsen, “WIMP Dark Matter and the QCD Equation of State,” Phys. Rev. D 71, 087302 (2005), [hep-ph/0501232].
  * [14] R. Sutrisno, R. Vilalta, and A. Renshaw, “A Machine Learning Approach for Dark-Matter Particle Identification Under Extreme Class Imbalance,” eds, ASP Conf. Ser., Vol. 523 (2018). https://www.uh.edu/~rvilalta/papers/2018/adass18.pdf
  * [15] P. Thakur, T. Malik, and T.K. Jha, “Towards Uncovering Dark Matter Effects on Neutron Star Properties: A Machine Learning Approach,” Particles 2024, 7, 80–95 (2024), arXiv:2401.07773 [hep-ph].
  * [16] C. D. Froggatt and H. B. Nielsen, “Standard model criticality prediction: Top mass 173 $\pm$ 5-GeV and Higgs mass 135 $\pm$ 9-GeV,” Phys. Lett. B 368, 96(1996), [hep-ph/9511371].
  * [17] C. D. Froggatt, H. B. Nielsen, and Y. Takanishi, “Standard model Higgs boson mass from borderline metastability of the vacuum,” Phys. Rev. D 64, 113014 (2001), [hep-ph/0104161].
  * [18] H. B. Nielsen, “PREdicted the Higgs Mass,” Bled Workshops Phys. 13 no. 2, 94–126, (2012), arXiv:1212.5716 [hep-ph].
  * [19] K. Kawana, “Multiple Point Principle of the Standard Model with Scalar Singlet Dark Matter and Right Handed Neutrinos,” PTEP, 023B04 (2015), arXiv:1411.2097 [hep-ph].
  * [20] K. Kawana, “Criticality and Inflation of the Gauged B-L Model,” PTEP, 073B04 (2015), arXiv:1501.04482 [hep-ph].
  * [21] Y. Hamada and K. Kawana, “Vanishing Higgs Potential in Minimal Dark Matter Models,” Phys. Lett. B 751, 164 (2015), arXiv:1506.06553 [hep-ph].
  * [22] N. Haba, H. Ishida, N. Okada, and Y. Yamaguchi, “Multiple-point principle with a scalar singlet extension of the Standard Model,” PTEP, 013B03 (2017), arXiv:1608.00087 [hep-ph].
  * [23] Y. Hamada, H. Kawai, K. Kawana, K.-y. Oda and K. Yagyu, “Gravitational waves in models with multicritical-point principle,” Eur. Phys. J. C 82 481 (2022), arXiv:2202.04221 [hep-ph].
  * [24] G.-C. Cho, C. Idegawa, and R. Sugihara, “A complex singlet extension of the Standard Model and Multi-critical Point Principle,“ Phys. lett. B 839 137757 (2023), arXiv:2212.13029 [hep-ph].
  * [25] H. Kawai, K. Kawana, K.-y. Oda, and K. Yagyu, “Quantum phase transition and absence of quadratic divergence in generalized quantum field theories,” arXiv:2307.11420 [hep-th].
  * [26] Y. Hamada, H. Kawai, K.-y. Oda, and K. Yagyu, “Dark matter in minimal dimensional transmutation with multicritical-point principle,” JHEP 01 087 (2021), arXiv:2008.08700 [hep-ph].
  * [27] Y. Hamada, H. Kawai, K. Kawana, K.-y. Oda, and K. Yagyu, “Minimal Scenario of Criticality for Electroweak Scale, Neutrino Masses, Dark Matter, and Inflation,” Eur. Phys.J. C 81 no. 11 962 (2021), arXiv:2102.04617 [hep-ph].
  * [28] H. Kawai and K. Kawana, “Multi-Critical Point Principle as the Origin of Classical Conformality and Its Generalizations,” PTEP 1, 013B11 (2022), arXiv:2107.10720 [hep-th].
  * [29] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori, and A. Strumia, “Higgs mass and vacuum stability in the Standard Model at NNLO,” JHEP 08 098 (2012), arXiv:1205.6497 [hep-ph].
  * [30] A. Andreas, et al., “Towards the next generation of simplified dark matter models,” Phys. Dark Universe 16, 49 (2017), arXiv:1607.06680 [hep-ex].
  * [31] S. R. Coleman and E. J. Weinberg,“Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,” Phys. Rev. D 7, 1888 (1973), [hep-th/0507214].
  * [32] D. Buttazzo et al.,“Investigating the near-criticality of the Higgs boson,” JHEP 12, 089 (2013), arXiv:1307.3536 [hep-th].
  * [33] Particle Data Group, R L Workman et al. ,“Review of Particle Physics,” PTEP, 083C01 (2022). https://academic.oup.com/ptep/article/2022/8/083C01/6651666