# Contextualized Word Embeddings Encode Aspects of Human-Like Word Sense
Knowledge

Sathvik Nair, Mahesh Srinivasan
Cognitive Science Program,
Department of Psychology,
University of California, Berkeley
Berkeley, CA, United States
<EMAIL_ADDRESS>
Stephan Meylan
Department of Brain and Cognitive Sciences,
Massachusetts Institute of Technology
Cambridge, MA, United States
<EMAIL_ADDRESS>
###### Abstract

Understanding context-dependent variation in word meanings is a key aspect of
human language comprehension supported by the lexicon. Lexicographic resources
(e.g., WordNet) capture only some of this context-dependent variation; for
example, they often do not encode how closely senses, or discretized word
meanings, are related to one another. Our work investigates whether recent
advances in NLP, specifically contextualized word embeddings, capture human-
like distinctions between English word senses, such as polysemy and homonymy.
We collect data from a behavioral, web-based experiment, in which participants
provide judgments of the relatedness of multiple WordNet senses of a word in a
two-dimensional spatial arrangement task. We find that participants’ judgments
of the relatedness between senses are correlated with distances between senses
in the BERT embedding space. Homonymous senses (e.g., bat as mammal vs. bat as
sports equipment) are reliably more distant from one another in the embedding
space than polysemous ones (e.g., chicken as animal vs. chicken as meat). Our
findings point towards the potential utility of continuous-space
representations of sense meanings.

##  1 Introduction

00footnotetext:  This work is licensed under a Creative Commons Attribution
4.0 International License. License details:
http://creativecommons.org/licenses/by/4.0/.

A key challenge in natural language understanding is grasping the range of
meanings that a word can take as a function of linguistic and non-linguistic
context. Successful linguistic comprehension involves constantly resolving
lexical ambiguity of this nature [frazier1990taking,
klepousniotou2002processing]. This re-use of word forms by speakers — relying
on listeners to choose the appropriate meaning depending on context — confers
language with higher communicative efficiency than it would otherwise have,
and may pose a solution to the problem of limited memory
[piantadosi2012communicative]. The re-use of word forms also allows speakers
to extend the lexicon to new communicative situations, for example to refer to
new objects, entities or processes using existing words in the language
[ramiro2018algorithms, xu2020conceptual, srinivasan2019children]. How these
context-specific meanings are understood and represented pose critical open
questions regarding the lexicon.

Following the conventions of lexicographic resources like natural language
dictionaries, variation in word meaning is often treated in a categorical
fashion: lexical types contain clusters of related meanings, or word senses.
Under this treatment, word tokens (instances of a word type that are used in
context) can be categorized into these word senses. Different sub-types of
lexical ambiguity are thus reflected in different relations between word
senses: pairs of word senses are polysemous if they are semantically related
[pustejovsky1998generative], or homonymous if this is not the case
[apresjan1974regular, tuggy1993ambiguity, lyons1995linguistic]. For example,
using bottle to refer to a container as well as to the liquid it contains is
an example of polysemy. By contrast, using bank to refer to a riverside or a
financial institution — two semantically unrelated meanings — constitutes
homonymy. Polysemous relations are often regular [apresjan1974regular], such
that many word types exhibit the same alternation (e.g., the container-for-
contents relation is also exemplified by box and glass); such patterns can
also be generalized to new words [srinivasan2019children].

Because of variation in the perceived relationship between senses across
language users, theories in psycholinguistics assert that humans treat
polysemy and homonymy as falling onto a continuous gradient
[tuggy1993ambiguity, crossley2010development]. This is corroborated by
psycholinguistic experiments which show that both adults and children
represent and process polysemous and homonymous senses differently
[frazier1990taking, rodd2004modelling, klepousniotou2008making,
rabagliati2013truth, macgregor2015sustained]. However, these theories of word
sense representation lie in stark contrast with many lexicographical
approaches, most notably WordNet [miller1990introduction]. Although WordNet
contains a vast store of word senses—and has historically been considered the
gold standard ontology of word senses [jurafsky2014speech]—it does not encode
relations among those senses, and thus does not distinguish between polysemy
and homonymy.

Models based on distributional semantics can help bridge the gap between
psychological theories of word senses and existing lexicographical resources.
Contextualized word embeddings (CWEs), especially BERT [devlin:2018], which is
derived from a Transformer-based architecture [vaswani2017attention], have the
potential to capture fine-grained distinctions in relatedness between word
senses, because they offer a continuous measure of relatedness between
individual uses of words in context [lake2020word]. While fine-tuned BERT
models are known to perform exceptionally well in word sense disambiguation
(WSD) tasks [wiedemann2019does, loureiro-jorge-2019-language, blevins2020wsd],
there are still open questions of whether BERT representations capture fine-
grained distinctions in sense relatedness.

In the present work, we evaluate whether contextualized embeddings from BERT
capture relationships between word senses similar to English speakers. We
collect human judgments of the relatedness among different senses of 32
English words through a web-based two-dimensional spatial arrangement task
[goldstone1994influences]. We then compare the experimental data to BERT
vectors for the same set of words. This is done through extracting and
analyzing BERT embeddings for word tokens in the Semcor corpus
[miller1993semantic], which has been annotated with word sense identifiers
from WordNet. Assessing whether relatedness in BERT representations
corresponds with human judgments of sense relatedness is an important test of
whether CWEs can be used to develop more realistic computational models of
word meanings.

##  2 Background

The distributional hypothesis proposes that a word’s meaning can be
represented by the lexical context in which it occurs
[harris1954distributional]. This insight underlies the success of
computational models that represent words as continuous-valued vectors
representing their surrounding lexical context. Latent semantic analysis
[dumais1988using], and neural models like Word2Vec[mikolovw2v] encode
information about all of a word’s senses in a single vector. By contrast,
newer models include contextualized embeddings: vector space representations
of specific word uses (tokens), reflecting the context of their use. These
richer word representations from contextualized embeddings have allowed
researchers to ask which aspects of linguistic knowledge are incorporated in
their vector spaces (See ?) for a review; see also ?)). For example, one of
the most notable discoveries from this line of work suggests that a sentence’s
dependency parse tree can be reconstructed from BERT embeddings
[hewitt2019structural, reif2019visualizing, jawahar-etal-2019-bert]. However,
despite progress investigating how BERT encodes syntactic information,
relatively little work has explored how it encodes semantic information such
as the relations between word senses.

While older models like Word2Vec may encode some aspects of word sense
[arora2018linear], we focus here on Transformer architectures
[vaswani2017attention], which may be able to find sense-distinguishing context
words at larger distances, and thus perform better for WSD. BERT-based
approaches have led to state of the art performance on word sense
disambiguation tasks [wiedemann2019does, loureiro-jorge-2019-language,
blevins2020wsd], but approaches analyzing the model’s representation of word
senses remain exploratory [reif2019visualizing, lake2020word]. Evidence from
?) shows that BERT embeddings for the same word in a large text corpus are
largely clustered based on their meanings, but the authors do not investigate
the model’s ability to encode canonically homonymous or polysemous
relationships. ?) note that BERT embeddings for the same word type vary as a
function of the position in the sentences they occur in. ?) assess BERT’s
capacity for homonym resolution by comparing relatedness of words that are
similar to a homonym in highly constraining contexts. ?) points out the need
for more psycholinguistic diagnostics of neural language models like BERT, and
compares BERT to human data from tasks such as commonsense inference. Our work
falls under this framework, focusing on comparing human judgements of word
sense relatedness to BERT representations.

##  3 Methods

We conducted a metalinguistic experiment (vs. a processing task testing
implicit knowledge) where participants used a web interface to assess the
relatedness of WordNet senses in a two-dimensional spatial arrangement task.
We then obtained BERT embeddings for word types in the Semcor corpus
[miller1993semantic] and compared them to the experimental data, looking at
both cosine distance in the embedding space as well as the accuracy of a sense
classifier using BERT’s contextualized word embeddings as input. All stimuli,
code and visualizations are available at https://osf.io/fm78w.

###  3.1 Data

We select a sample of word types for analysis from Semcor
[miller1993semantic], which has WordNet sense annotations for 235,000 tokens
from the Brown corpus [francis1979brown]. We use the corpus reader from the
Python Natural Language Toolkit (NLTK) [bird-loper-2004-nltk] to access the
corpus. The fact that syntax is represented within BERT embeddings
[hewitt2019structural] implies that BERT embeddings can easily capture
distinctions in part of speech, so we focus on lexical ambiguity within part-
of speech (specifically nouns and verbs) as a more challenging test case.

For the behavioral experiment, we selected word types across a range of sense
entropy values (i.e., uniform to highly peaked sense distributions). We
created a multinomial distribution over the senses of each lemma, i.e., (word
type, part of speech) pair, and computed its entropy as follows:
$\displaystyle-\sum_{s\in L}\frac{c_{s}}{c_{L}}\log(\frac{c_{s}}{c_{L}})$,
where $L$ is a lemma, $s$ is a sense and $c$ corresponds to a frequency, or
count in the corpus. We removed stopwords and lemmas with zero entropy (79.3%
of all lemmas). This yielded 444 word types.

For experimental stimuli, we chose 32 word types. 20 word types came from
Semcor, including 11 low/medium entropy word types and 9 high entropy word
types. We defined high entropy to be greater than 1.5 when rounded to the
nearest tenth, and medium/low entropy to be less than this value, based on the
distribution of sense entropy in the available Semcor data (in Figure 1). To
account for variability in participants’ placement of tokens in the spatial
arrangement task, we needed to normalize measurements in the interface on a
per-trial basis, so we selected words with three senses or more. We chose word
types with varying entropy values to determine if this quantity had an effect
on the correlation between BERT vectors and human judgements; relatedness
between embeddings for words with more unpredictable sense distributions may
be less consistent with results from the experiment. To avoid overwhelming
participants, we selected words with fewer than eight senses. We also chose
six additional words of theoretical importance that exemplify patterns of
regular polysemy that have been observed across languages from ?), and
manually chose six words with three senses, one of which was less semantically
related to the other two. Because we expected these words to exhibit the
greatest differences in pairwise similarity, we elicited judgments from all
participants on these items. More details about how these stimuli were
presented and exclusion criteria are in Appendix B.

####  3.1.1 Participants

105 undergraduate students from a major research university participated in
the experiment and were compensated with class credit. Upon providing informed
consent, participants reported their experience with English and other
languages to ensure that data were collected from proficient English speakers
(defined as at least 50% of daily language use); one participant was excluded
using this criterion.

###  3.2 Experiment

From the set of 32 word types, each participant received 14 word types as
stimuli. We used a two-dimensional spatial-arrangement task
[goldstone1994influences], because it allowed us to efficiently capture
psychological judgments, and because tasks of this nature effectively capture
relatedness in a high-dimensional semantic space [richie2020spatial].
Participants were told to place less related sense tokens further apart from
one another on the canvas, and more closely related tokens near each other.
The experimental interface is shown in Appendix A. To place the sense tokens,
participants were given definitions (from WordNet) and example sentences (from
Semcor; in some cases shortened for brevity). The task was untimed, and
participants were encouraged to introspect for as long as necessary about the
meanings of the presented senses before making their placements. Partcipants
were encouraged to adjust their placements to reflect all senses.

###  3.3 Relatedness Matrices

For each word, we collected spatial (x,y) positions of each of the word’s
senses from each participant. To derive an estimate of the relatedness of
senses for each word type, we normalized the distances such that they were
rescaled according to the largest reported pairwise distance. This controlled
for variation in the absolute amount of space in the canvas that different
participants used when making their placements. After excluding participants
with unreliable responses (Appendix B), we then averaged relatedness matrices
for each word type across the remaining participants. This yielded a single
aggregate relatedness matrix for each word type which could then be directly
compared with model results for a word type. To account for the larger number
of participants reporting data for shared stimuli, we selected data from a
random set of participants whose size corresponded to the average amount of
test items ($n$ = 29). Data from a minimum of 21 participants and a maximum of
37 participants were used to construct the aggregate relatedness matrix for
each word type.

###  3.4 Modeling

We retrieved BERT embeddings for labeled tokens from the Semcor corpus
[miller1993semantic] and ran classifiers to distinguish between different
senses of individual words based on BERT vectors. We compared these data to
results from the experiment based on two metrics: distance in the embedding
space and classification accuracy, assessing their similarity through
correlation [Hirst06evaluatingwordnet-based].

Given a word type and a part of speech, we derived word embeddings from BERT
as follows. For each sense of the (word type, part of speech) pair, we
retrieved sentences from Semcor, and tokenized them using rules specified by
the BERT authors [devlin:2018]. We loaded a pre-trained BERT model BERT-base
from ?) and ran the forward pass on each sentence, extracting the activations
corresponding to the type, and storing the summed activations of the final
four layers, which has produced strong results in word sense disambiguation
[loureiro-jorge-2019-language].

###  3.5 Word Sense Classifier

As a matter of due diligence, we first evaluated the effectiveness of BERT
representations for word sense disambiguation (classification) in the Semcor
corpus. While WSD-focused work has used the Semcor dataset for training and
tested on other datasets [raganato2017word], we confirmed that BERT
demonstrates similar levels of performance on Semcor. To this end, we assessed
performance of multiclass logistic regression models to predict word senses
for 401 word types from Semcor. Senses were omitted from this classification
whenever there were fewer than 10 tokens. Each logistic regression model was
trained to predict WordNet sense labels for each instance of a sense based on
its contextual word embedding from BERT. We conducted 5-fold cross validation
and applied L1 regularization to prevent overfitting. To confirm consistency
with prior work on WSD [loureiro-jorge-2019-language, reif2019visualizing,
blevins2020wsd], we report the average F1 score (weighted for each sense)
across the runs for each word type. Comparable F1 scores relative to this
prior work would suggest that it is appropriate to further investigate the
geometry implicit in these contextual word embeddings.

####  3.5.1 Extracting Cosine Distances Between Word Sense Centroids

After qualitatively reproducing previous word sense disambiguation results
using BERT embeddings, we then tested whether the relationship between BERT
embeddings of senses parallels human judgments of relatedness between senses.
Because BERT generates one vector for each use of a word token, we computed
the centroids of the BERT embeddings corresponding to each sense, and then
compared different senses by using the cosine distance of these centroids to
one another. In a broad range of models, cosine distances between vectors
corresponding to word types can encode the degree of semantic relatedness
[dumais1988using, mikolovw2v, joulin2016fasttext], so we can use this same
metric with BERT embeddings to compare word senses to one another. Following
the same procedure as the relatedness matrices, we define relatedness as (1 -
cosine distance), such that the largest cosine distance (least related pair)
takes the value of 0 and the smallest cosine distance (most closely related
pair) takes the value of 1 for each word type. We stored these relatedness
measures between sense centroids in a matrix analogous to the aggregate
relatedness matrices from the behavioral experiment. To evaluate their fit to
human data, we computed the Spearman rank correlation coefficient between the
upper triangular entries in these two distance matrices. A nonparametric
measure of correlation is appropriate given that the relationship is not
necessarily linear.

##  4 Results

First, we verified that BERT can discriminate between senses in Semcor
comparably to other test datasets. We then compared relatedness estimates
derived from distances in the BERT embedding space to the aggregate
relatedness measures from the experiment. We then conduct two exploratory
analyses, the first comparing the relatedness of homonymous and polysemous
sense pairs, and the second examining the utility of pairwise sense confusion
as a measure of relatedness.

###  4.1 Classification Results for All Semcor Data

We compared the distribution of F1 scores of the classifiers on the 401 words
from Semcor (BERT failed to process 43 of the words) with two baselines:
random choice and selecting the most frequent sense (Figure 1).

Method | All Types in Semcor | Types in Behavioral Expt.
---|---|---
Number of Available Types | 401 | 32
Random Sense | 0.480 | 0.423
Majority Sense | 0.441 | 0.403
Logistic Regression on BERT Embeddings | 0.757 | 0.797
Table 1: Average F1 scores (across types and train-test splits) for BERT-based
word sense disambiguation in the Semcor corpus.

Classification performance on all 401 words, including 25 words used in the
behavioral experiment, were similar to those reported by ?), ?), and Reif et
al. ?) (0.739, 0.754, and 0.711, respectively), which used test sets from ?).
We find that type-wise F1 score is negatively correlated with sense entropy
computed from Semcor ($\beta_{sense\\_entropy}=-0.074,R^{2}=0.06$; Fig. 1).

Figure 1: Classifier performance for word types with respect to their sense
entropy in Semcor. F1 score for each type is averaged over cross validation
folds and weighted by sense frequency. Vertical line represents boundary
between low/medium and high entropy word types.

###  4.2 Comparing Human Relatedness with BERT Word Sense Geometry

We computed the Spearman’s rank correlation coefficient between human
judgments of sense relatedness (from the aggregate relatedness matrices) and
measures of word sense relatedness from the centroids of BERT contextualized
word embeddings. This yielded a positive correlation (Spearman’s
$r=0.565,p<0.001$), with a 95% confidence interval of (0.459, 0.655). To
confirm that this correlation was above that expected by random chance, we
compared it to a random baseline established by 1000 draws of randomly
generated sense placements for 29 participants (the average number of
participants per item). This correlation was much lower (Spearman’s
$r=0.062,p<0.001$, 95% CI: -0.065 - 0.208). We report the correlation between
relatedness measures from the embedding space and human judgements by part of
speech and high vs. low/medium entropy in Table 2. Correlations between human
relatedness judgments and BERT relatedness were substantially higher among low
to medium than high entropy words(Table 2). We also find that correlations
among human and model relatedness are higher among verbs than nouns.

####  4.2.1 Polysemous vs. Homonymous Sense Pairs

We conducted an exploratory analysis to see if human relatedness judgments
differed between homonymous and polysemous relations, and whether the geometry
of BERT embeddings captured this distinction between word senses. In the
absence of gold-standard datasets labeling polysemous and homonymous word
sense relationships,111We could in principle use natural language
dictionaries, some of which list polysemous and homonymous senses under
separate entries. However, such definitions would not necessarily map to
WordNet labeled data. we tagged pairs of senses as polysemous or homonymous
ourselves for the set of words in the behavioral experiment. For human
judgments, we took the set of average distances between polysemous and
homonymous pairs of senses. For BERT embeddings, we took the cosine distances
between pairs of polysemous and homonymous sense centroids.

The density plots in Figure 2 show that participants in the behavioral
experiment judged the polysemous pairs to be more similar (i.e., less distant)
than homonymous pairs (Mann-Whitney $U=516,p<0.001$). BERT-based
representations reproduced this basic pattern (Mann-Whitney $U=493,p<0.001$),
though many relatedness estimates were more dispersed for both polysemous and
homonymous pairs.

Figure 2: Distribution of distances between pairs of senses from the human
data (left) and from BERT (right), across word types.

To illustrate the differences between how BERT represents polysemous and
homonymous relationships, we show t-SNE and dendrogram visualizations in
Figure 3. For table.n (Figure 3, top), we show that two senses related by
polysemy (piece of furniture vs. a tablesetting) are judged to be closer
together than homonymous pairs (either of the first two senses vs. “a set of
data arranged in rows and columns”). For cover.v, all sense pairs were
polysemous. Instances of the same sense were still closer to one another in
the embedding space, but tokens corresponding to different senses were much
less clearly distinguished except for cover.v.04, which may be well-separated
from other senses because it refers to a metaphorical meaning of the lemma. We
find that the BERT centroids’ cosine distances and the aggregated relatedness
judgements are strongly correlated (Spearman’s $r=0.851,p<0.001$), with a 95%
confidence interval of (0.638, 0.943).

Figure 3: Visualizations of example BERT embeddings for stimuli table.n
cover.n. Left column contains results of t-SNE dimensionality reduction with
small points indicating word tokens and large points indicating sense
centroids. Right column contains results of single linkage agglomerative
clustering.

###  4.3 Predicting Human Relatedness Judgments with Pairwise Confusion

One possibility is that the cosine distance computed over all dimensions of
the BERT embedding space overlooks the possibility that some dimensions may be
more or less useful for discriminating word senses depending on the word type.
One alternative is to use the pairwise confusion probabilities from multiclass
logistic regression (described in Section 3.5) as a measurement of pairwise
relatedness in the model. This choice invokes substantive theoretical
questions: in principle, sense relatedness and discriminability could be
orthogonal. For example, two senses could be judged as closely related by
humans, but they could in principle be able to discriminate between instances
of those senses without errors. In the absence of human sense discrimination
performance data, we leave this question to future work. Nonetheless, we
investigated the utility of pairwise sense confusion as an alternative
predictor for human judgments of sense relatedness.

To this end, we summed the confusion matrices when senses from the stimuli are
classified during each iteration of cross-validation from Section 4.1. We
normalized each item in the matrix by the number of true labels, such that it
represents the probability an item was predicted given its true class. Across
all word types with available data (20 out of 32 types, including 150 out of
189 sense pairs), we found a positive Spearman’s rank correlation between
entries in the confusion matrices and matrices of the corresponding
relatedness judgements $(r=0.649,p<0.001)$, with a 95% confidence interval of
(0.592, 0.7). For this measure, we considered all entries in both matrices, as
the confusion matrices are not necessarily symmetric. Because this analysis
omits word senses with fewer than 10 tokens in Semcor, this evaluation
reflects a smaller set of items than the one reported in Section 4.2 for
cosine-based relatedness. When we computed the correlation between cosine-
based relatedness and human aggregated relatedness matrices on this smaller
set, the correlation remains lower (Spearman’s $r=0.518,p<0.001$), with a 95%
confidence interval of (0.391, 0.627). This suggests that pairwise confusion
in BERT may be a stronger predictor of human relatedness judgments. We also
report correlations stratified by part of speech and type sense entropy in
Table 2.

Metric | Part of Speech |  Word Sense Entropy
---|---|---
| Nouns | Verbs  | High | Low/Medium
Relatedness (1 - Normalized Cosine Distance)  | 0.487 | 0.623  | 0.345 | 0.570
Pairwise Confusion  | 0.609 | 0.627  | 0.518 | 0.731
Table 2: Spearman rank correlations between relatedness judgements and
confusion matrices in BERT embedding space, split over part of speech and
entropy level

To illustrate the approach, we present aggregate relatedness matrices and
confusion matrices for the word type area.n in Fig. 4. In this case, the
experimental data (Fig 4, A) was more closely aligned with the confusion
matrix (C) the than the cosine similarity of centroids (B).

(a) Subfigure 1 list of figures text

(b) Subfigure 2 list of figures text

(c) Subfigure 3 list of figures text

(d) Subfigure 4 list of figures text

Figure 4: Mean human relatedness judgements, cosine similarity matrix, and
confusion matrix for senses of area.n.

We also evaluated sense classification accuracy on the basis of the homonymous
and polysemous sense pairs from the preceding exploratory analysis. Among
homonymous sense pairs, the classifiers achieved an average F1 score of 0.992.
Among the polysemous sense pairs, the classifier achieved an average F1 score
of 0.752.

##  5 Discussion

We investigated the ability of an artificial neural network model that
represents word tokens with contextualized word embeddings to capture human-
like relations among English word senses. On a subset of word types from the
Semcor corpus, we reproduced previous word sense disambiguation results. We
then showed that these same BERT embeddings capture a significant amount of
information regarding the relationship between word senses, and are able to at
least partially reproduce human relatedness judgments. An exploratory analysis
revealed that BERT-based measures of the relatedness between pairs of
homonymous senses are much lower than for pairs of polysemous senses, matching
human intuitions for the same set of senses. Pairwise confusion from the sense
classifiers provided a slightly better predictor of human judgments of sense
relatedness compared to distance in the embedding space, although this may be
unsurprising given that pairwise confusion reflects some additional degree of
supervision. Analyzing the error rates of classifiers trained on BERT
embeddings of the stimuli, we also provided evidence that their results are
more accurate for homonymous sense pairs compared to polysemous sense pairs.

Following in the approach of comparing BERT vectors to experimental data from
human participants [ettinger2020bert], the present research addresses how BERT
represents word senses, bridging the gap between human and computational
models of lexical ambiguity resolution. Recent progress showing BERT’s
performance on word sense disambiguation [wiedemann2019does, loureiro-
jorge-2019-language, blevins2020wsd] indicates that BERT-based models perform
better than past approaches, and our work specifies that the representations
they use to accomplish this are relatively consistent with human intuitions.
Existing exploratory work aims to analyze how BERT represents word senses
[reif2019visualizing, lake2020word], but we systematically evaluate these
claims with both WordNet senses and human data. Our work corroborates claims
by ?) that BERT captures relations between homonymous senses. Indeed, poor
performance in discriminating polysemous pairs of words senses suggests that
more work needs to be done on capturing polysemous relations; one possibility
is to explore how BERT could be fine-tuned to capture relations exemplified by
regular polysemy. One limitation from these findings is that the dataset only
covers 32 word types, which we hope to address in future work.

More generally, our findings suggest that a continuous measure of sense
relatedness derived from neural language models could potentially be used to
augment existing lexicographic resources like WordNet, in this case providing
a measure of relatedness between senses. We encourage other researchers
developing sense ontologies, especially those found through word sense
induction, to consider representing the relationships among word senses in
addition to discovering new sense inventories.

##  6 Conclusion

Through comparing results from a behavioral experiment with data from
contextualized word embeddings, we demonstrate that new Transformer-based
neural network architectures may reflect human intuitions about the
relationships between word senses, at least in English. Exploratory analyses
suggest that these measures of relatedness between sense pairs reflect a
distinction between polysemous and homonymous relationships between word
senses, but that these models are much more effective in discriminating
homonymous sense pairs than polysemous ones. By demonstrating basic levels of
consistency with human judgements, we hope to stimulate further research that
combines contextualized word embedding techniques with discrete, symbolic
ontologies to develop a more cognitively informed model of the lexicon.

## Acknowledgements

We would like to thank Steven Piantadosi, Ruthe Foushee, Sammy Floyd, and
Jessica Mankewitz for thoughtful discussion of this work, and Jon Wehry for
ensuring the experiment ran smoothly. We also thank the reviewers for helpful
feedback, and the workshop organizers for giving us the opportunity to share
this work.

##  Appendix A Experiment Details

Figure 5: A sample trial in the experiment

##  Appendix B Trial Types and Exclusion Criteria

The first two of the eighteen trials participants received were presented as
training trials to ensure that participants were familiar with the interface;
data from these trials were discarded. Participants received a mixture of
shared trials completed by all participants, and test trials where only a
subset of participants provided judgments. The inclusion of both trial types
gave us a consistent set of word types for which all participants contributed
data (allowing us to evaluate which participants gave unreliable responses
compared to all other participants), while at the same time characterizing a
broad set of word types with more sparse responses. Test trials were drawn
from the set of 26 lemmas consisting of words from ?). To identify
participants who provided low-quality data, we computed hold-one-out
correlations for each participant using their relatedness matrices for the
shared trials, and excluded participants whose data had a rank correlation
with the hold-one-out averages that was lower than 0.4. This threshold
corresponds roughly to the 92nd percentile of scores if sense tokens are
placed randomly in the interface.

In addition to the shared and test trials, participants also saw two repeat
trials, drawn from the same set as the test trials, to evaluate the
reliability of their responses within the same testing session. Participants
with rank correlations in reported distances between their original and repeat
trials lower than 0.2 were excluded from analysis, corresponding to the 70th
percentile of correlations if sense tokens are placed randomly (this threshold
was lower because of variation in both the number of senses and the recency of
the test trial). Ten participants who failed to meet both criteria were
excluded from further analysis.

##  Appendix C Experimental Stimuli

Word Type | Entropy | No. of Senses | Freq. in Semcor | Part of Speech [head to column names]expt_semcor_types.csv
---|---|---|---|---
Table 3: All experimental stimuli from Semcor, with entropy, part of speech, and number of senses Word Type | Source | No. of Senses | Freq. in Semcor | Part of Speech
---|---|---|---|---
book.n | ?) | 3 | 58 | Noun
glass.n | 3 | 38 | Noun
face.n | 4 | 142 | Noun
heart.n | 3 | 60 | Noun
door.n | 3 | 135 | Noun
school.n | 3 | 143 | Noun
table.n | Shared Stimuli | 3 | 81 | Noun
plane.n | 3 | 39 | Noun
right.n | 3 | 54 | Noun
degree.n | 3 | 53 | Noun
model.n | 3 | 36 | Noun
foot.n | 3 | 146 | Noun
Table 4: Stimuli selected from ?) and Shared Stimuli (manually selected).


#  $SU(5)_{L}\times U(1)_{Y}$ electroweak unification

Adrian Palcu  "Aurel Vlaicu" University of Arad,
2 Elena Drăgoi Street, Arad-310330, Romania

###### Abstract

We propose here - for the first time in the literature, to our best knowledge
- an electroweak unification based on the $SU(5)_{L}\times U(1)_{Y}$ gauge
group. The spontaneous symmetry breaking takes place in the manner
$SU(5)_{L}\times U(1)_{Y}\rightarrow U(1)_{em}$, due to a particular Higgs
sector consisting of five scalar quintuplets. Each scalar quintuplet acquires
its own vacuum expectation value, by means of a proper parametrization which
is worked out once the overall vacuum expectation value in the model is
established. The decoupling of the low energy regime (corresponding to the
Standard Model) from the high scale (required by out model here) is
straightforwardly achieved in order to preserve the consistency with the
present experimental data. Finally, a promising phenomenological outcome is
derived by simply tuning a single free parameter. Our results include, besides
a viable one-parameter mass spectrum, also the prediction of precisely three
generations in the fermion sector and the electric charge quantization.

##  1 Introduction

In a recent paper [1], the author took a step further in working out some
consequences of the "general Cotăescu method" (gCm) conceived for treating
$SU(3)_{c}\times SU(n)_{L}\times U(1)_{Y}$ gauge models [2], by proving that
this particular approach can precisely predict the electric charge
quantization observed in nature. At the same time, a promising
phenomenological outcome can be achieved, once the free parameters necessary
to split the scalar sector underpinning a coherent Higgs mechanism are
properly set. To this end, the model must undergo a spontaneous symmetry
breaking (SSB) in its electroweak sector up to the universal $U(1)_{em}$
group, while the renormalization requirement is kept valid for all the model’s
sectors throughout the proceedings. The main result - that allows for a
plethora of new degrees of freedom beyond those coming from the Standard Model
(SM) - regards the mass spectrum of the model. In our approach, the low energy
regime can be satisfactorily decoupled from the overall vacuum expectation
value (VEV) of such general gauge models, in order not to compromise the SM
phenomenology (constructed within the well-known $SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}$ gauge group [3]-[5]). In this connection, a
compelling argument was sketched - see section 5 in Ref.[1] \- for the most
general case when $n$ is arbitrary. When particular values of $n$ in the gCm
are taken into consideration, our approach leads to some concrete extensions
of the SM. Not more than three such possible extensions [1], as the strict
renormalization condition informs us. So, a vast amount of work has been
published to date dealing with the so called 3-3-1 gauge models [6]-[32] or
3-4-1 gauge models [33]-[57], but none approaching 3-5-1 model, the last one
allowed by the gCm.

But, why extending the SM? The reason to extend the SM is fueled by some
recent experimental challenges - regarding, among others, neutrino
oscillations or dark matter puzzle - that have definitely no explanation at
the level of the SM. Instead, they seem to be promisingly addressed within the
richer framework supplied by such extended gauge models. Not to add certain
theoretical devices that are natural in such SM-extensions and aim at
elucidating some issues (that already have come of age), such as the strong-CP
problem, the electric charge quantization, the precise number of three fermion
generations, the tiny (seesaw) neutrino masses accounting for the oscillation
phenomenon - to name but a few. Most plausible candidates for dark matter and
most suitable ingredients for various types of seesaw mechanism can be
definitely found in a larger extension of the SM. That is why we address here
the task of constructing the 3-5-1 model (with no exotic electric charges)
based on the $SU(3)_{c}\times SU(5)_{L}\times U(1)_{Y}$ gauge group. The
theoretical structure will be rigorously built assuming the gCm formalism is
combined with the restrictive _asymptotic freedom_ requirement of the QCD [58,
59]. Afterwards, the consistency with the available experimental data will be
highlighted. Our strategy here is to apply and effectively exploit all the
prescriptions of the gCm for the particular case $n=5$.

The paper is organized as follows. In sections 2 \- 4 the three sectors
(fermion, gauge, scalar) of the model are presented. In section 5 the electric
charge quantization is obtained, while section 6 is devoted to some
phenomenological results regarding the mass spectrum and some rough numerical
estimates depending on the overall breaking scale in the model. A
phenomenological outlook and related comments are stated in the final section
7. Not to omit two Appendixes, where some mathematical details regarding the
Lie algebra involved in the procedings and the diagonalization of the mass
matrix are displayed.

##  2 Fermion sector

We consider the simplest class of 3-5-1 gauge models, namely the one with no
other electric charges than $0$, $\pm$ e, and make use of the prescriptions
obtained by the author in the general case [1]. Due to the asymptotic freedom
imposed by the color $SU(3)$ symmetry of the QCD, there must be precisely
three fermion generations in order to cancel the chiral anomalies that could
occur in the electroweak sector. Therefore, the fermion sector will consist of
the following left-handed (lower index L) representations (here, the generic L
stands for leptons, generic Q stands for quarks, and $i=1,2,3$):

$L_{iL}=\left(\begin{array}[]{c}N_{i}^{\prime\prime}\\\ N_{i}^{\prime}\\\
N_{i}\\\ \nu_{i}\\\ e_{i}\end{array}\right)_{L},\quad
Q_{1L}=\left(\begin{array}[]{c}U^{\prime\prime}\\\ U^{\prime}\\\ U\\\ u\\\
d\end{array}\right)_{L},\quad
Q_{2L,3L}=\left(\begin{array}[]{c}D_{2,3}^{\prime\prime}\\\
D_{2,3}^{\prime}\\\ D_{2,3}\\\ d_{2,3}\\\ u_{2,3}\end{array}\right)_{L}$ (1)

along with their corresponding singlet right-handed partners. The capital
letters on the first three positions in each generation are assigned to new
fermions (specific to this 3-5-1 model), whilst the small letters denote the
well-known SM fermions. For the moment, their electric charges are just
supposed to be those observed in nature, but this will be rigorously proved
only in section 5.

According to the gCm, for $n=5$ \- from eqs.(35)-(37) in Ref.[1] \- a
renormalizable model of this kind must have the fermion representations
transforming under the $SU(3)_{c}\times SU(5)_{L}\times U(1)_{Y}$ gauge group
in the manner:

$L_{iL}\sim(\boldsymbol{1},\boldsymbol{5},-\frac{1}{5})\quad,\quad
e_{iR}(\boldsymbol{1},\boldsymbol{1},-1)\quad,\quad\nu_{iR},N_{iR},N_{iR}^{\prime},N_{iR}^{\prime\prime}\sim(\boldsymbol{1},\boldsymbol{1},0)$
(2)

$Q_{1L}\sim(\boldsymbol{3},\boldsymbol{5},\frac{7}{15})\quad,\quad
Q_{kL}\sim(\boldsymbol{3},\boldsymbol{5^{*}},-\frac{2}{15})$ (3)

$u_{kR},u_{R},U_{R},U_{R}^{\prime},U_{R}^{\prime\prime}\sim(\boldsymbol{3},\boldsymbol{1},\frac{2}{3})\quad,\quad
d_{R},D_{kR},D_{kR}^{\prime},D_{kR}^{\prime\prime}\sim(\boldsymbol{3},\boldsymbol{1},-\frac{1}{3})$
(4)

with $k=2,3$.

The above assignments are sufficient in order to get a renormalizable 3-5-1
gauge model for they cancel all the chiral anomalies that could plague
effective results. In addition, they will play a crucial role in the electric
charge quantization.

##  3 Gauge sector

While the QCD remains an exact $SU(3)_{c}$ vector symmetry, the Lie algebra of
the electroweak $SU(5)_{L}\times U(1)_{Y}$ gauge group provides us with a
large set of gauge interactions. In this connection, besides the SM
interactions due to neutral (diagonal) bosons $A_{em}$ (massless) and $Z$
($91.1876\pm 0.0021$ GeV [60]), three more neutral weak interactions arise as
coupled to heavier (also diagonal, Hermitian) bosons
$Z^{\prime}$,$Z^{\prime\prime}$ and $Z^{\prime\prime\prime}$, respectively. At
the same time, there are 18 more off-diagonal (non-Hermitian) entries in the
adjoint representation, other than $W^{\pm}$ ($80.379\pm 0.012$ GeV [60]).
Some of these 18 are charged, some are neutral - as it will be made clear
further on. The electroweak boson adjoint representation stands as

$A_{\mu}=\left(\begin{array}[]{ccccccccc}D_{\mu}^{1}&&Y_{\mu}^{\prime\prime
0}&&Y_{\mu}^{\prime 0}&&Y_{\mu}^{0}&&Y_{\mu}^{\prime\prime+}\\\ \\\
Y_{\mu}^{\prime\prime 0*}&&D_{\mu}^{2}&&X_{\mu}^{\prime\prime
0}&&X_{\mu}^{\prime 0}&&Y_{\mu}^{\prime+}\\\ \\\ Y_{\mu}^{\prime
0*}&&X_{\mu}^{\prime\prime 0*}&&D_{\mu}^{3}&&X_{\mu}^{0}&&Y_{\mu}^{+}\\\ \\\
Y_{\mu}^{0*}&&X_{\mu}^{\prime 0*}&&X_{\mu}^{0*}&&D_{\mu}^{4}&&W_{\mu}^{+}\\\
\\\
Y_{\mu}^{\prime\prime-}&&Y_{\mu}^{\prime-}&&Y_{\mu}^{-}&&W_{\mu}^{-}&&D_{\mu}^{5}\end{array}\right)$
(5)

where the diagonal entries are in order:

$\begin{array}[]{ll}D_{\mu}^{1}=&\frac{1}{2}A_{\mu}^{3}+\frac{1}{2\sqrt{3}}A_{\mu}^{8}+\frac{1}{2\sqrt{6}}A_{\mu}^{15}+\frac{1}{2\sqrt{10}}A_{\mu}^{24}+YB^{0}_{\mu}\\\
\\\
D_{\mu}^{2}=&-\frac{1}{2}A_{\mu}^{3}+\frac{1}{2\sqrt{3}}A_{\mu}^{8}+\frac{1}{2\sqrt{6}}A_{\mu}^{15}+\frac{1}{2\sqrt{10}}A_{\mu}^{24}+YB^{0}_{\mu}\\\
\\\
D_{\mu}^{3}=&-\frac{1}{\sqrt{3}}A_{\mu}^{8}+\frac{1}{2\sqrt{6}}A_{\mu}^{15}+\frac{1}{2\sqrt{10}}A_{\mu}^{24}+YB^{0}_{\mu}\\\
\\\
D_{\mu}^{4}=&-\frac{3}{2\sqrt{6}}A_{\mu}^{15}+\frac{1}{2\sqrt{10}}A_{\mu}^{24}+YB^{0}_{\mu}\\\
\\\ D_{\mu}^{5}=&-\frac{2}{\sqrt{10}}A_{\mu}^{24}+YB^{0}_{\mu}\end{array}$ (6)

The off-diagonal entires are generically denoted as
$B_{\mu}^{\alpha\beta}=\frac{1}{\sqrt{2}}(A_{\mu}^{\alpha}\pm
iA_{\mu}^{\beta})$ with $\alpha,\beta=1,2,3,4,5$, $\alpha\neq\beta$. Some of
the off-diagonal entries could correspond to charged bosons (if either
$\alpha=5$ or $\beta=5$), but also to neutral bosons as well (if
simultaneously $\alpha\neq 5$ and $\beta\neq 5$) - as one will be able to
infer, once the charge quantization will be performed in the fermion sector.
However, even at this stage, one can state (based only on algebraical
arguments) that $B_{\mu}^{\alpha\beta}=B_{\mu}^{*\beta\alpha}$.

##  4 Scalar sector

In order for the model to achieve its SSB, there are needed no less than five
quintuplets, each quintuplet acquiring its own VEV, according to a proper
prametrization [1, 2] that will ensure the VEVs split. Before the SSB, the
scalar quintuplets transform like

$\phi^{(i)}=\left(\begin{array}[]{l}\phi_{1}^{(i)}\\\ \phi_{2}^{(i)}\\\
\phi_{3}^{(i)}\\\ \phi_{4}^{(i)}\\\
\phi_{5}^{(i)}\end{array}\right)\sim(\boldsymbol{1},\boldsymbol{5},Y_{i})\qquad
i=1,2,\ldots,5$ (7)

with the generic $Y_{i}$ the would-be hypercharge associated to $U(1)_{Y}$. As
the gCm restricts [2], these would-be hypercharges must obey Tr($Y$)=0 for the
scalar multiplets, namely $Y_{1}+Y_{2}+Y_{3}+Y_{4}+Y_{5}=0$.

After the SSB, each scalar quintuplet will exhibit its own VEV

$\left\langle\phi^{(i)}\right\rangle=\left(\begin{array}[]{c}\vdots\\\
\eta_{i}\left\langle\varphi\right\rangle\\\ \vdots\end{array}\right)\qquad
i=1,2,\ldots,5$ (8)

but at least one or two of them must range in the hundreds of GeV to recover
the SM phenomenology and prevent it from being altered by the high energy
scale in the model. The remaining VEVs, of course, are ranging higher and they
will depend on the free parameters $\eta_{i}$. The general solution for a good
renormalizable model predicts - eqs. (33)-(34) in Ref.[1] \- precisely for the
fifth quintuplet $Y_{5}=\frac{4}{5}$ and for the first four ones
$Y_{k}=-\frac{1}{5}$, $k=1,2,\ldots,4$.

Now, what are the parameters $\eta_{i}$ and why do we need them? They are key
ingredients in the gCm - section 4 in Ref.[2] \- in order to split the VEVs in
the manner $\eta_{i}\left\langle\varphi\right\rangle$ for each
$i=1,2,\ldots,n$, once the overall VEV in the model
$\left\langle\varphi\right\rangle$ is established via the minimum condition
imposed to the renormalizable scalar potential allowed by the gauge symmetry
(without any other supplementary restrictions). Obviously $\eta_{i}\in(0,1)$.
The parameter matrix involved in the scalar sector looks like

$\eta=\left(\sqrt{1-\eta_{0}^{2}}\right)\textrm{Diag}(\eta_{1},\eta_{2},\eta_{3},\eta_{4},\eta_{5})$
(9)

with $\eta_{0}$ the corresponding coefficient of $\varphi\sim$ ($\bf{1}$,
$\bf{1}$, 0) in the kinetic term of the potential (eq.(30) in Ref.[2]).
Actually, this will be the unique scalar field predicted by the gCM that, in
turn, will split into the five surviving Higgses $h_{i}=\eta_{i}\varphi$ ,
after SSB takes place.

The compulsory trace condition $Tr(\eta^{2})=1-\eta_{0}^{2}$ is simply
achieved as a sum of the parameters

$\eta_{1}^{2}+\eta_{2}^{2}+\eta_{3}^{2}+\eta_{4}^{2}+\eta_{5}^{2}=1$ (10)

Hence, we have

$\left\langle\phi_{1}\right\rangle^{2}+\left\langle\phi_{2}\right\rangle^{2}+\left\langle\phi_{3}\right\rangle^{2}+\left\langle\phi_{4}\right\rangle^{2}+\left\langle\phi_{5}\right\rangle^{2}=\left\langle\varphi\right\rangle^{2}$
(11)

In our particular 3-5-1 case, we propose, for the sake of simplicity, the
following plausible parametrization

$\left\langle\phi_{1}\right\rangle\simeq\left\langle\phi_{2}\right\rangle\simeq\left\langle\phi_{3}\right\rangle=\sqrt{\frac{1-a}{3}}\left\langle\varphi\right\rangle$
(12)

$\left\langle\phi_{4}\right\rangle=\sqrt{\frac{a-b}{2}}\left\langle\varphi\right\rangle\quad,\quad\left\langle\phi_{5}\right\rangle=\sqrt{\frac{a+b}{2}}\left\langle\varphi\right\rangle$
(13)

that, obviously, satisfies eq.(10).

At the end of the proceedings, the desired phenomenology is obtained if one
tunes the parameters to be small enough, i.e. $a,b\rightarrow 0$. Under these
circumstances, the quite degenerate VEVs
$\left\langle\phi_{1}\right\rangle\simeq\left\langle\phi_{2}\right\rangle\simeq\left\langle\phi_{3}\right\rangle$
range similarly to the overall VEV $\left\langle\varphi\right\rangle$, while
$\left\langle\phi_{4}\right\rangle$ and $\left\langle\phi_{5}\right\rangle$
are responsible for the low energy regime (precisely corresponding to the SM
phenomenology). The assumption of the strict equality for the first three VEVs
is taken just for simplicity. At any time one can perform some small
deviations from it in order to split the three VEVs, but this will not shift
them from the high energy regime. It will only add some complications in the
high energy regime, due to the mixing among heavier degrees of freedom that
will arise.

##  5 Electric charge quantization

The expression of the electric charge operator established in the general
method (eq.(58) in Ref.[2]) was worked out by the author in the case of
general models with no exotic electric charges (eq.(9) in Ref.[1]). Hence, for
our particular 3-5-1 model of interest here, the electric charge operator,
when considering $n=5$, yields:

$Q_{em}^{\rho}=g\sin\theta_{W}\left[T_{24}^{\rho}\frac{\sin\theta}{\sin\theta_{W}}+\frac{g^{\prime}}{g}Y^{\rho}\frac{\cos\theta}{\sin\theta_{W}}\right]$
(14)

where $\varrho$ labels each representation of the gauge group. It will take
the particular form

$\frac{Q_{em}^{\rho}}{e}=\frac{1}{2\sqrt{10}}\left(\begin{array}[]{ccccc}1\\\
&1\\\ &&1\\\ &&&1\\\
&&&&-4\end{array}\right)\frac{\sin\theta}{\sin\theta_{W}}+\frac{g^{\prime}}{g}Y^{\rho}\frac{\cos\theta}{\sin\theta_{W}}$
(15)

Once the relation between the angles $\theta$ (of the gCm) and $\theta_{W}$
(of the SM) is established - via eq.(10) in Ref.[1] \- it gets its 3-5-1
specific form, namely

$\sin\theta=2\sqrt{\frac{2}{5}}\sin\theta_{W}$ (16)

and supplies the following formula for the couplings match

$g=g^{\prime}\frac{\sqrt{5}\sin\theta_{W}}{\sqrt{5-8\sin^{2}\theta_{W}}}$ (17)

Consequently, the final expression of the electric charge operator can be
easily inferred by combining the previous three equations. It becomes:

$\frac{Q_{em}^{\rho}}{e}=2\sqrt{\frac{2}{5}}T_{24}^{\rho}+Y^{\rho}$ (18)

The elementary electric charge now can be obtained in terms of coupling
constants of the groups involved in the electroweak unification

$\frac{\sqrt{5}g^{\prime}g}{\sqrt{5g^{2}+8g^{\prime 2}}}=e$ (19)

Under these conditions, the electric charge operator eq.(18) leads
straightforwardly to the observed quatization for each representation in eqs.
(2)-(4), once their would-be hypercharges are plugged in. For the sake of
completeness, we display below the eigenvalues of the electric charge operator
for all fermion representationsthat are in order:

$Q_{em}^{(\boldsymbol{5},-\frac{1}{5})}=\left(\begin{array}[]{ccccc}0\\\ &0\\\
&&0\\\ &&&0\\\ &&&&-1\end{array}\right)$ (20)

for the lepton sector, and

$Q_{em}^{(\boldsymbol{5},\frac{7}{15})}=\left(\begin{array}[]{ccccc}2/3\\\
&2/3\\\ &&2/3\\\ &&&2/3\\\ &&&&-1/3\end{array}\right)$ (21)

$Q_{em}^{(\boldsymbol{5^{*}},-\frac{2}{15})}=\left(\begin{array}[]{ccccc}-1/3\\\
&-1/3\\\ &&-1/3\\\ &&&-1/3\\\ &&&&2/3\end{array}\right)$ (22)

for the quark sector, respectively.

The trace condition for the electric charge operator
$\sum_{\varrho}\textrm{Tr}(Q^{\varrho})=0$ in any good phenomenological model
is fulfilled by this particular outcome, as one can easily check out. For a
viable phenomenology it is obvious that the first three entries in each
fermion quintuplet must be much heavier than their partners on the 4th and 5th
position, since the latter ones are the well-known SM fermions. However, the
fermion mass spectrum will be utterly obtained once a proper Yukawa sector is
employed, but this lies beyond the scope of our paper.

##  6 Boson mass spectrum

The next step towards a good phenomenological model is to obtain a viable mass
spectrum for the boson sector. The latter - with the above resulting quantized
charges - evidently, will consist only of neutral and singly-charged physical
bosons.

According to the general prescription in Ref.[2], the off-diagonal entries in
eq.(5) aquire the following masses:

$M_{j}^{i}=\frac{g}{2}\sqrt{1-\eta_{0}^{2}}\left\langle\varphi\right\rangle\sqrt{\eta_{i}^{2}+\eta_{j}^{2}}$
(23)

The lighter of them (corresponding to the 4-5 position)

$M^{2}(W_{\mu}^{\pm})=\frac{1}{4}g^{2}(1-\eta_{0}^{2})\left\langle\varphi\right\rangle^{2}a$
(24)

corresponds to the well-known $W^{\pm}$ from the SM. In order to simplify
expressions, one can adopt the notation
$m^{2}=\frac{1}{4}g^{2}(1-\eta_{0}^{2})\left\langle\varphi\right\rangle^{2}$.
Furthermore, the other bosons masses are:

$M^{2}(Y_{\mu}^{\pm},Y_{\mu}^{\prime\pm},Y_{\mu}^{\prime\prime\pm})=m^{2}\left(\frac{1-a}{3}+\frac{a+b}{2}\right)=\frac{M^{2}(W_{\mu}^{\pm})}{a}\left(\frac{1}{3}+\frac{a+3b}{6}\right)$
(25)

$M^{2}(Y{}_{\mu}^{0},Y_{\mu}^{\prime 0},Y_{\mu}^{\prime\prime
0})=m^{2}\left(\frac{1-a}{3}+\frac{a-b}{2}\right)=\frac{M^{2}(W_{\mu}^{\pm})}{a}\left(\frac{1}{3}+\frac{a-3b}{6}\right)$
(26)

$M^{2}(X_{\mu}^{0},X_{\mu}^{\prime 0},X_{\mu}^{\prime\prime
0})=m^{2}\left(\frac{2-a}{3}\right)=\frac{M^{2}(W_{\mu}^{\pm})}{a}\left(\frac{2}{3}\right)(1-a)$
(27)

For the diagonal (Hermitian) bosons, i.e. the neutral weak mediators
$Z,Z^{\prime},Z^{\prime\prime},Z^{\prime\prime\prime}$, the gCm -
eqs.(53)-(54) in Ref.[2] \- provides us with the mass matrix:

$M_{\hat{i}\hat{j}}^{2}=\left\langle\varphi\right\rangle^{2}Tr(B_{\hat{i}}B_{\hat{j}})$
(28)

where

$B_{\hat{i}}=g\left[D_{\hat{i}}+\nu_{\hat{i}}(D\nu)\frac{1-\cos\theta}{\cos\theta}\right]\eta$
(29)

with $\hat{i},\hat{j}$ labeling the diagonal generators of the Cartan
subalgebra, _i.e._ $\hat{i}=1,...,4$, in our particular case. Versors
$\nu_{\hat{i}}=\nu^{\hat{i}}$ with $\nu_{\hat{i}}\nu^{\hat{i}}=1$ are
associated to the orthogonal directions on the electromagnetic direction in
the parameter space. Around $\nu$ is performed the $SO(n)$ rotation of angle
$\theta$ that separates the massless photon (see eqs. (48)-(49) in [2]) from
the massive $Z$s. The choice of these versors actually discriminates among
different models. As we work in the no-exotic-electric-charges model, the
versor choice is obviously $\nu_{n^{2}-1}=1$, while all the others vanish
$\nu_{\hat{i}}=0$ for $\hat{i}=1,2,3$. Actually, we have already exploited
this choice when writing down the electric charge operator for no-exotic
models.

Under these circumstances, the diagonal entries of the neutral bosons matrix
read

$M_{11}^{2}=M_{22}^{2}=m^{2}\left(\frac{2}{3}\right)(1-a)$ (30)

$M_{33}^{2}=\frac{m^{2}}{6}\left(1+\frac{7}{2}a-\frac{9}{2}b\right)$ (31)

$M_{44}^{2}=\frac{m^{2}}{10\cos^{2}\theta}\left(1+\frac{15}{2}a+\frac{15}{2}b\right)$
(32)

For the first two neutral bosons one gets - as expected - no mixing. The
masses of the heavier $Z^{\prime\prime}$ and $Z^{\prime\prime\prime}$ are thus
degenerate.

$M_{12}^{2}\simeq M_{21}^{2}\simeq M_{13}^{2}\simeq M_{31}^{2}\simeq
M_{23}^{2}\simeq M_{32}^{2}=0$ (33)

There is however one exception that occurs between the last two bosons, $Z$
and $Z^{\prime}$, that notwithstanding mix with one another.

$M_{34}^{2}=M_{43}^{2}\simeq\frac{m^{2}}{2\sqrt{15}\cos\theta}\left(1-\frac{5}{2}a+\frac{3}{2}b\right)$
(34)

Therefore, one must perform the diagonalization proceedure on the matrix
$M^{2}(Z,Z^{\prime})$ that has the following expression:

$M^{2}(Z,Z^{\prime})=\frac{m^{2}}{2}\left(\begin{array}[]{cc}\frac{1}{3}\left(1+\frac{7}{2}a-\frac{9}{2}b\right)&\frac{1}{\sqrt{15}\cos\theta}\left(1-\frac{5}{2}a+\frac{3}{2}b\right)\\\
\\\
\frac{1}{\sqrt{15}\cos\theta}\left(1-\frac{5}{2}a+\frac{3}{2}b\right)&\frac{1}{5\cos^{2}\theta}\left(1+\frac{15}{2}a+\frac{15}{2}b\right)\end{array}\right)$
(35)

By imposing that one of the eigenvalues of this symmetric matrix must be

$M^{2}(Z)=\frac{M^{2}(W_{\mu}^{\pm})}{\cos^{2}\theta_{W}}=\frac{m^{2}a}{\cos^{2}\theta_{W}}$
(36)

in accordance with the established SM condition, and exploiting that in our
3-5-1 model $\cos\theta=\sqrt{1-\frac{8}{5}\sin^{2}\theta_{W}}$ holds, one
gets through the diagonalization procedure (see Appendix) the following
relation between parameters

$b=-a\tan^{2}\theta_{W}$ (37)

Consequently, the mass of the $Z^{\prime}$ will be

$M^{2}(Z^{\prime})=m^{2}\left[\frac{4}{3}\left(\frac{1-\sin^{2}\theta_{W}}{5-8\sin^{2}\theta_{W}}\right)-\frac{a}{3}\frac{(5-10\sin^{2}\theta_{W}-4\sin^{4}\theta_{W})}{(5-8\sin^{2}\theta_{W})(1-\sin^{2}\theta_{W})}\right]$
(38)

At this point, the full boson mass spectrum depending on the unique free
parameter $a$ is fully determined. It remains only to be tuned in order to
keep consistency with the experimental data. With the above formulas - eqs
(25)-(27), (30) and (38) - the analytic expressions will take the shape:

$M(Y_{\mu}^{\pm},Y_{\mu}^{\prime\pm},Y_{\mu}^{\prime\prime\pm})=\frac{M(W_{\mu}^{\pm})}{\sqrt{a}}\sqrt{\frac{1}{3}+a\left(\frac{1-3\tan^{2}\theta_{W}}{6}\right)}$
(39)

$M(Y{}_{\mu}^{0},Y_{\mu}^{\prime 0},Y_{\mu}^{\prime\prime
0})=\frac{M(W_{\mu}^{\pm})}{\sqrt{a}}\sqrt{\frac{1}{3}+a\left(\frac{1+3\tan^{2}\theta_{W}}{6}\right)}$
(40)

$M(X_{\mu}^{0},X_{\mu}^{\prime 0},X_{\mu}^{\prime\prime
0},Z^{\prime\prime},Z^{\prime\prime\prime})=\frac{M(W_{\mu}^{\pm})}{\sqrt{a}}\sqrt{\left(\frac{2}{3}\right)(1-a)}$
(41)

$M(Z^{\prime})=\frac{M(W_{\mu}^{\pm})}{\sqrt{a}}\sqrt{(0.322-0.343a)}$ (42)

Table 1: Boson masses at various breaking scales (in TeV) mass\ $\left\langle\varphi\right\rangle$ | 1TeV | 2TeV | 7TeV | 10TeV | 13TeV | 14TeV
---|---|---|---|---|---|---
$W^{\pm}$ | 80.38GeV | 80.38GeV | 80.38GeV | 80.38GeV | 80.38GeV | 80.38GeV
$Z^{0}$ | 91.18GeV | 91.18GeV | 91.18GeV | 91.18GeV | 91.18GeV | 91.18GeV
$Z^{\prime 0}$ | 180.2GeV | 368.3GeV | 1.297TeV | 1.854TeV | 2.452TeV | 2.653TeV
$Y^{\pm},Y^{\prime\pm},Y^{\prime\prime\pm}$ | 189.8GeV | 377.7GeV | 1.320TeV | 1.886TeV | 2.452TeV | 2.653TeV
$Y^{0},Y^{\prime 0},Y^{\prime\prime 0}$ | 194.6GeV | 380.2GeV | 1.320TeV | 1.886TeV | 2.452TeV | 2.653TeV
$X^{0},X^{\prime 0},X^{\prime\prime 0}$ | 259.9GeV | 530.3GeV | 1.867TeV | 2.668TeV | 3.469TeV | 3.740TeV
$Z^{\prime\prime 0},Z^{\prime\prime\prime 0}$ | 259.9GeV | 530.3GeV | 1.867TeV | 2.668TeV | 3.469TeV | 3.740TeV

Some numerical estimates for the boson masses - at various breaking scales
defined by the parameter $a$ \- are inferred and displayed in Table 1.

To each $a\in(0,1)$ there is a corresponding breaking scale in the model,
according to
$\left\langle\varphi\right\rangle_{SM}=\sqrt{a}\left\langle\varphi\right\rangle$.
This is straightforwardly obtained if one simply equates the mass of the
$W^{\pm}$ boson, both in the SM and in our 3-5-1 model, with
$\left\langle\varphi\right\rangle_{SM}=246$GeV.

At the same time, one can plot the boson mass spectrum (virtually, for all
values of the parameter $a$). We confine however, in Figure 1, to displaying
it only for a particular domain from $(0,1)$ where the high scale is clearly
decoupled from the low energy regime. This region favors, as expected, very
small $a$.

Figure 1: Boson masses depending on the free parameter a

##  7 Summary and outlook

In this article we set up a plausible electroweak unification by taking into
consideration for that purpose the particular $SU(5)_{L}\times U(1)_{Y}$ gauge
group. Within this framework, we applied rigorously the prescriptions and
results of the the theoretical device conceived by Cotăescu for
$SU(n)_{L}\times U(1)_{Y}$ gauge models with SSB [1, 2]. Once the model’s
symmetry was broken up to the universal electromagnetic $U(1)_{em}$, the
method makes room for a viable boson mass spectrum by tuning a single free
parameter $a$. Among the appealing features inherited from the general method,
we have here the exact predictions for the number of fermion generations and
the electric charge quantization.

Henceforth, one can work out some more phenomenological consequences that our
3-5-1 model is suitable for. We briefly review some of them here:

  * •

Weak currents. The main next step in exploiting our theoretical approach from
a phenomenological standpoint, is to determine the exact couplings of all the
weak currents - properly speaking, to get the quantization of the "neutral
charges" $Q^{\rho}(Z_{\hat{i}})$. That is, one has to compute the following
diagonal operators (eq.(59) in Ref.[2])

$Q^{\rho}(Z_{\hat{i}})=g\left[D_{\hat{k}}^{\rho}-\nu_{\hat{k}}\left(D^{\varrho}\nu\right)(1-\cos\theta)-\nu_{\hat{k}}\frac{g^{\prime}}{g}Y^{\rho}\sin\theta\right]\omega_{\cdot\hat{i}}^{\hat{k}\cdot}$
(43)

for the particular 3-5-1 model at hand. Moreover, the eigenvalues of these
operators must be obtained. Based on the resulting weak charges, a huge amount
of phenomenology can be computed and put eventually under test at LHC and
future facilities.

To this end, one must first determine the $SO(4)$ rotation that properly
diagonalizes the mass matrix (35). In our model, assuming the degenerate
heavier $Z^{\prime\prime}$ and $Z^{\prime\prime\prime}$ bosons, it can be put
as

$\omega=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&\cos\phi&\sin\phi\\\ 0&0&-\sin\phi&\cos\phi\end{array}\right)$ (44)

Under these circumstances, the neutral currents in the 3-5-1 model, read:

$Q^{\rho}(Z)=g\left[T_{15}^{\rho}\cos\phi+\left(T_{24}^{\varrho}\cos\theta-\frac{g^{\prime}}{g}Y^{\rho}\sin\theta\right)\sin\phi\right]$
(45)

$Q^{\rho}(Z^{\prime})=g\left[-T_{15}^{\rho}\sin\phi+\left(T_{24}^{\varrho}\cos\theta-\frac{g^{\prime}}{g}Y^{\rho}\sin\theta\right)\cos\phi\right]$
(46)

$Q^{\rho}(Z^{\prime\prime})=gT_{8}=g\frac{1}{2\sqrt{3}}\textrm{Diag}(1,1,-2,0,0)$
(47)

$Q^{\rho}(Z^{\prime\prime\prime})=gT_{3}=g\frac{1}{2}\textrm{Diag}(1,-1,0,0,0)$
(48)

Obviously, when inspecting eqs. (45)-(48), one observes that SM fermions
couple only with $Z$ and $Z^{\prime}$ being, thus, sterile with respect to the
two heavier $Z^{\prime\prime}$ and $Z^{\prime\prime\prime}$. This is a good
phenomenological outcome since, to date, the occurrence of $Z^{\prime\prime}$
and $Z^{\prime\prime\prime}$ has not been reported at the available energies
the present facilities can supply. A detailed determination of all the
couplings remains be performed elsewhere [61], since it lies beyond the scope
of our present paper. Yet, we can state that all the couplings of the ordinary
particles (denoted by small letters leptons and quarks in (1)-(4)) recover -
by computing (45) - precisely their SM-couplings. So, our SM-extension doesn’t
alter at all the SM phenomenology.

  * •

Neutrino mass. The phenomenon of neutrino oscillation (observed for the first
time by Kamiokande [62] and SNO [63] collaborations) requires tiny neutrino
masses for the three $\nu_{iL}$. As long as a fermion mass term presumes the
existence of both left-handed and right-handed chiral components, the task of
directly generating such non-zero Dirac masses for neutrinos is unbearable
within the SM, since no $\nu_{iR}$ can be found therein. Even if they would
have been added by hand, a non natural fine tuning of their Yukawa couplings
must definitely be imposed. Among many proposals for a way out of this
conundrum, some seesaw mechanisms [64]-[70] were conceived to generate tiny
Majorana masses for $\nu_{iL}$, but still resorting to an extra-SM hypothesis:
the existence of some sterile heavy right-handed Majorana neutrinos ($N_{iR}$)
violating the $B-L$ symmetry at a scale much higher than the SM scale.
Meanwhile, many versions of the seesaw mechanism have been unfolded. Our model
is flexible enough in order to supply many ingredients to such a goal and
accommodate a lot of seesaw mechanisms. First of all, since all right-handed
neutrinos in our model are singlets belongng to the representation $({\bf
1},{\bf 1},0)$ any of them can be removed from the spectrum without altering
the cancellation of the chiral anomaly necessary to preserve the
renormalizability of the model. This allows one to conceive various seesaw
scenarios. Some of the massive neutral particles included in lepton
quintuplets may acquire their Majorana mass through an effective dimension-5
operator approach if one takes into consideration such terms

$-\frac{Y^{ij}}{\Lambda}\overline{L}L(\phi_{i}\times\phi_{j})$ (49)

with $i,j=1,2,3$. However, even the radiative mass generation scenarios are
not excluded in the 3-5-1 framework, but all these deserves a distinct study
in its own right.

  * •

Dark Matter. The cumbersome issue of the presence of the Dark Matter in the
universe can be addressed within this kind of models, since there is a set of
several particles (fermions, scalars, even neutral gauge bosons) that are
sterile when it comes to the ordinary matter. Therefore they could play the
role of WIMP particles and their relic abundance in the universe can, in
principle, be estimated once the overall breaking scale is established. In
particular 3-3-1 models there are several attempts to propose DM candidates
[71]-[79]. If one proceeds with our simple 3-5-1 model in which the three
heavier breaking scales are degenerate, for instance all the three degenerate
heavy neutral fermions could be suitable cold DM candidates. A detailed
discussion in this respect deserves to be separately made too.

  * •

Higgs sector. It is evident that the scalar sector of such 3-5-1 model are
compounded by a plethora of scalar degrees of freedom, five of the neutral
ones developing their proper VEVs and becoming Higgs particles to be hunted at
LHC. A detailed analysis of the scalar spectrum could supply interesting
results with regard to their masses and couplings. In the case of 3-3-1 models
with no exotic electric charges, the author succeded in proving that only two
physical Higgses survive the SSB [18, 32], while the third one comes out
massless and sterile (with no couplings at all). Some of them could also be
put forward as plausible DM candidates.

##  Appendix A Lie algebra $su(5)_{L}\times u(1)_{Y}$

We briefly present the unitary irreducible representations (irreps) and
algebraical structure of the semi-simple $SU(5)\times U(1)$ Lie group with a
focus on its Hermitian generators, of interest in constructing the adjoint
representation defining the gauge bosons of the model.

For $U(1)_{Y}$ group things are simple, as its unitary irreps are constructed
just as phase factors of its associated Lie algebra, once the single Hermitian
generator is identified with $I$ and $g^{\prime}$ stands as its coupling
constant. There is one corresponding real parameter in the algebra, asociated
to the real gauge field $B^{0}_{\mu}$. Therefore, the $U(1)$ unitary irreps
can be put as

$U(\xi^{0})=\exp{(-ig^{\prime}YB^{0}_{\mu}(x))}$ (50)

For $SU(5)_{L}$ things are a little more complicated due to the fact that the
Hermitian generators $T^{\dagger}_{a}=T_{a}$ of its Lie algebra $su(5)$ do not
commute with one another. That is, they are subject to the canonical
commutation relations of the form

$\left[T_{a},T_{b}\right]=iC_{ab}^{c}T_{c}$ (51)

with $C_{ab}^{c}$ standing for the real and completely antisymmetric structure
constants. For each $a,b,c,...$ there is a corresponding Hermitian generator
out of the $n^{2}-1=24$ such Hermitian generators. Out of the $24$, only $4$
are diagonal (Cartan sub-algebra) and $20$ are off-diagonal.

The trace condition (that holds for every representation $\rho$) implies the
normalization of the generators in the manner

$\text{Tr}(T_{a})=0\implies\text{Tr}(T_{a}T_{a})=\frac{1}{2}\delta_{ab}$ (52)

The $SU(5)$ unitary irreducible representations can be put as

$U(\xi)=\exp{(-igT_{a}A^{a}_{\mu}(x))}$ (53)

where $g$ is the coupling constant assumed, throughout our proceedings here,
to be identical with the SM coupling of $SU(2)_{L}$

Finally, we display the concrete expressions of the four diagonal Hermitian
generators that are manipulated in our computations. They are in order:
$D_{1}=T_{3}$, $D_{2}=T_{8}$, $D_{3}=T_{15}$, $D_{4}=T_{24}$.

$T_{3}=\frac{1}{2}\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&-1&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\end{array}\right)\quad,\quad
T_{8}=\frac{1}{2\sqrt{3}}\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&-2&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\end{array}\right)$ (54)

$T_{15}=\frac{1}{2\sqrt{6}}\left(\begin{array}[]{ccccc}1&0&0&0&0\\\
0&1&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&-1&0\\\ 0&0&0&0&0\end{array}\right)\quad,\quad
T_{24}=\frac{1}{2\sqrt{10}}\left(\begin{array}[]{ccccc}1&0&0&0&0\\\
0&1&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&-4\end{array}\right)$ (55)

The $SU(5)$ Hermitian generators of the group transform according to the
adjoint representation of the $su(5)$ algebra

$U(\xi)T_{a}U^{\dagger}(\xi)=\text{Adj}(\xi)^{b}_{a}T_{b}$ (56)

where

$\text{Adj}(\xi)=\exp(-ig\text{adj}(\xi))$ (57)

and

$\text{adj}(\xi)=\xi^{a}\text{adj}(T_{a})$ (58)

which is exactly the adjoint representation of the algebra

$\text{adj}(T_{a})^{b}_{c}=iC_{ac}^{b}$ (59)

We conclude our algebraical detour here with the manner the fermion
representation transform under the gauge group of interest here. Denoting by
$\psi^{\rho}$ any fermion multiplet (belonging to the unitary irreducible
representation $\rho$ of the $SU(5)_{L}\times U(1)_{Y}$ gauge group), we have
the transformation rule:

$\psi^{\rho}(x)\rightarrow\exp[-i(gA^{\rho}_{\mu}(x)+g^{\prime}Y^{\rho}B_{\mu}(x))]\psi^{\rho}(x)$
(60)

in order to obey the gauge symmetry of the model. The gauge fields transform
under the same gauge group in the manner:

$A_{\mu}(x)\rightarrow
U(\xi)A_{\mu}(x)U^{\dagger}(\xi)-\frac{i}{g}(\partial_{\mu}U(\xi))U^{\dagger}(\xi)\quad,\quad
B_{\mu}(x)\rightarrow B_{\mu}(x)+\partial_{\mu}\xi^{0}(x)$ (61)

##  Appendix B Diagonalization of the boson mass matrix

Here we enter into some technical details regarding the eigenvalues of the
mass matrix (35). We start with the characteristic equation that reads

$\left|\begin{array}[]{cc}\frac{1}{3}\left(1+\frac{7}{2}a-\frac{9}{2}b\right)-\frac{2a}{1-sin^{2}\theta_{W}}&\frac{1}{\sqrt{15}\cos\theta}\left(1-\frac{5}{2}a+\frac{3}{2}b\right)\\\
\\\
\frac{1}{\sqrt{15}\cos\theta}\left(1-\frac{5}{2}a+\frac{3}{2}b\right)&\frac{1}{5\cos^{2}\theta}\left(1+\frac{15}{2}a+\frac{15}{2}b\right)-\frac{2a}{1-sin^{2}\theta_{W}}\end{array}\right|=0$
(62)

once it is enforced the well-known SM-condition
$M^{2}(Z)=M^{2}(W^{\pm})/cos^{2}\theta_{W}$. This restriction will lead to a
particular relation between the two free parameters - $a$ and $b$ \- as the
determinant is developed. It reduces to simply summing the following
coeffcients of the powers of the parameters

  * •

free term:
$\frac{1}{3}\left(\frac{1}{5\cos^{2}\theta}\right)-\left(\frac{1}{\sqrt{15}\cos\theta}\right)\left(\frac{1}{\sqrt{15}\cos\theta}\right)=0$

  * •

$a$:
$-\frac{(5+7\sin^{2}\theta_{W})}{2(1-\sin^{2}\theta_{W})}-\frac{(5-17\sin^{2}\theta_{W})}{2(1-\sin^{2}\theta_{W})}+\frac{10(1-\sin^{2}\theta_{W})}{2(1-\sin^{2}\theta_{W})}=0$

  * •

$b$: $-\frac{9}{2}+\frac{15}{2}-2\frac{3}{2}=0$

  * •

$ab$:
$\frac{15(5+7\sin^{2}\theta_{W})}{4(1-\sin^{2}\theta_{W})}+\frac{9(5-17\sin^{2}\theta_{W})}{4(1-\sin^{2}\theta_{W})}+\frac{30(1-\sin^{2}\theta_{W})}{4(1-\sin^{2}\theta_{W})}=-\frac{-72\sin^{2}\theta_{W}}{1-\sin^{2}\theta_{W}}$

  * •

$a^{2}$:
$\frac{(5+7\sin^{2}\theta_{W})(5-17\sin^{2}\theta_{W})}{4(1-\sin^{2}\theta_{W})^{2}}-\frac{25}{4}=-\frac{-36\sin^{4}\theta_{W}}{(1-\sin^{2}\theta_{W})^{2}}$

  * •

$b^{2}$: $-15\left(\frac{9}{4}\right)-\frac{9}{4}=-36$

Now, the characteristic equation becomes

$\left(a\tan^{2}\theta_{W}+b\right)^{2}=0$ (63)

and it ensures, up to a factor $\frac{m^{2}}{2}$, the following one-parameter
mass matrix

$\left(\begin{array}[]{cc}\frac{1}{3}\left[1+a\frac{(7+2\sin^{2}\theta_{W})}{2(1-\sin^{2}\theta_{W})}\right]&\frac{1}{\sqrt{3(5-8\sin^{2}\theta_{W})}}\left[1-a\frac{(5-2\sin^{2}\theta_{W})}{2(1-\sin^{2}\theta_{W})}\right]\\\
\\\
\frac{1}{\sqrt{3(5-8\sin^{2}\theta_{W})}}\left[1-a\frac{(5-2\sin^{2}\theta_{W})}{2(1-\sin^{2}\theta_{W})}\right]&\frac{1}{5-8\sin^{2}\theta_{W}}\left[1+a\frac{(15-30\sin^{2}\theta_{W})}{2(1-\sin^{2}\theta_{W})}\right]\end{array}\right)$
(64)

once $b$ was replaced by $-a\tan^{2}\theta_{W}$ in eq.(35). The trace of this
matrix will provide us with its eigenvalues $M_{1}=M^{2}(Z)$ and
$M_{2}=M^{2}(Z^{\prime})$ where
$M^{2}(Z^{\prime})=Tr(M^{2})-\frac{m^{2}a}{1-\sin^{2}\theta_{W}}$ and hence
the concrete expression in eq.(38).

It remains only to be determined the appropriate rotation matrix ($\omega$)
that does the job of diagonalization, since it is a key ingredient in
determining fermions’ couplings to neutral bosons. However, this lies beyond
the scope of our article here.

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