# CST5: Data Augmentation for Code-Switched Semantic Parsing

Anmol Agarwal, Jigar Gupta, Rahul Goel, Shyam Upadhyay
Pankaj Joshi, Rengarajan Aravamudhan
Google Assistant
<EMAIL_ADDRESS>This work
was done during an internship at Google

###### Abstract

Extending semantic parsers to code-switched input has been a challenging
problem, primarily due to a lack of supervised training data. In this work, we
introduce CST5, a new data augmentation technique that finetunes a T5 model
using a small seed set ($\approx$100 utterances) to generate code-switched
utterances from English utterances. We show that CST5 generates high quality
code-switched data, both intrinsically (per human evaluation) and
extrinsically by comparing baseline models which are trained without data
augmentation to models which are trained with augmented data. Empirically we
observe that using CST5, one can achieve the same semantic parsing performance
by using up to 20x less labeled data. To aid further research in this area, we
are also releasing (a) Hinglish-TOP, the largest human annotated code-switched
semantic parsing dataset to date, containing 10k human annotated Hindi-English
(Hinglish) code-switched utterances, and (b) Over 170K CST5 generated code-
switched utterances from the TOPv2 dataset. Human evaluation shows that both
the human annotated data as well as the CST5 generated data is of good
quality.

##  1 Introduction

Code-switching (CS) occurs when a speaker alternates between two or more
languages within a single conversation. With the increasing ubiquity of voice-
based assistants (e.g., Google Assistant, Alexa), the ability to parse code-
switched utterances has become one of the key challenges towards building
multilingual conversational agents.

Unfortunately, majority of semantic parsing datasets are in English, with very
few datasets exhibiting code-switching. While data collection efforts
Einolghozati et al. (2021); Mehnaz et al. (2021) have been made to bridge this
gap, collecting such datasets requires time-consuming and expensive human
annotations from raters who are proficient in multiple languages, making it
difficult to obtain large datasets. Another line of work generates artificial
code-switched data, either using parallel sentences in the two languages as
supervision Winata et al. (2019) or learning a generative language model from
large code-switched corpora Chang et al. (2019). However, these approaches
assume the availability of large corpora (>50k sentences), either parallel,
monolingual or code-switched in the languages of interest, which may not be
available for the domains of interest.

Original English Sentence |  CST5 Generated Code Switched Utterance
---|---
Create a new alarm for 9AM on monday 18th June |  Monday 18th June ko subah 9 bajhe ke liye ek naye alarm ko create karen
Remove Jim from my reminder to party next wednesday |  Agle wednesday ko party ke liye Jim ko mere reminder se hata den
Table 1: A few examples of code switched data generated by CST5. Even with 100
seed examples, the model generates good quality data across multiple domains.
English and Hindi tokens are color coded.  Figure 1: An overview of our
approach. We first use human annotators to annotate utterances. We use these
to fine tune a mT5 model for code switching and then use the rest of the
monolingual corpus to generate more code switched data.  Figure 2: We finetune
mT5 on parallel human-annotated data and do inference using the finetuned mt5
model using the monolingual data

In this paper we propose CST5, a strategy to obtain code-switched labeled data
by generating code switched utterances from a English semantic parsing
dataset, using only a small ($\approx$100 examples) seed set of (English,
Code-switched) example pairs to learn the transformation. This allows us to
effectively leverage existing semantic parsing datasets in English to derive
supervision for code-switched semantic parsing. Using CST5, we achieve an
average 25% improvement on the exact match (EM) accuracy, equivalent to having
a order of magnitude more training data. A few examples of the CS utterances
produced by CST5 are shown in Table 1.

Our work is inspired by the growing popularity of using large pre-trained
language models (LM) (such as BERT Devlin et al. (2018), T5 Raffel et al.
(2019) etc.) as a means of data augmentation (Kumar et al., 2019; Anaby-Tavor
et al., 2020; Lee et al., 2021, inter alia). Particularly, CST5 relies on mT5
Xue et al. (2020), a large pre-trained multilingual LM, which is fine-tuned to
perform code-switching using the small seed set, and then applied on a large
monolingual semantic parsing dataset in English to generate code switched
utterances. By mapping the labeled semantic parse for the English utterance,
one can recover the semantic parse to the code-switched utterance, thus
obtaining a labeled code-switched semantic parsing dataset. We observe that
the mT5 model is well suited to the task of code-switching and the generated
code-switched utterances were deemed of high quality upon manual inspection by
human raters, judged to be over 89% semantically equivalent to the original
English utterance, and 98% natural.

We use CST5 to generate synthetic code switched data and measure the effect of
data augmentation on parser performance with varying seed sizes and across
various semantic parsing domains. We observe a 20x data reduction for code
switching when using the XXL model. Even with a 100 seed examples, our
approach matches the performance of a model trained with 2000 examples for the
XXL mT5 parser. To summarize, in this work:

  * •

We introduce a novel data augmentation technique to generate synthetic code
switched data for semantic parsing. Using a small seed set of parallel data,
the technique converts mono-lingual data to code switched data.

  * •

We did additional human evaluation of the generated data showing high data
quality.

  * •

We conducted experiments on TOPv2 as well as CSTOPEinolghozati et al. (2021)
showing significant gains in EM accuracy.

  * •

We release a code-switched task-oriented semantic parsing dataset, containing
over 10k Hindi + English human annotated CS utterances (Hinglish-TOP) along
with over 170K synthetically generated CS utterances and their corresponding
semantic parse.

##  2 Related Work

Our work is at the intersection of two areas in NLP: data augmentation for
code switching and conversational semantic parsing.

###  2.1 Data Augmentation for Code Switching

Data augmentation is seen as a flexible and model independent tool for
performance improvement by synthetically generating training data for the task
of interest (Jia and Liang, 2016; Andreas, 2019; Akyürek et al., 2020, inter
alia). We discuss some approaches for synthetically generating code switched
data below.

Winata et al. (2019) generated code switched utterances using parallel
sentences in the two languages as supervision. Chang et al. (2019) proposed an
unsupervised method to generate code-switching sentences from monolingual
sentences using GANs. Linguistically motivated data augmentation has also been
explored in Lee et al. (2019) and Pratapa et al. (2018). Such techniques
require expert knowledge or large amounts of parallel data which might be hard
to get in a new domain. Instead our work uses small seed sets to generate
synthetic data which is more scalable.

##### Language Model based Data Augmentation

Kumar et al. (2021) used pre-trained transformer models, like BART Lewis et
al. (2020) and GPT-Radford et al. (2019), for general data augmentation. Our
work leverages large pretrained language models (LMs) Raffel et al. (2019);
Brown et al. (2020) to generate new synthetic examples. Our approach draws
inspiration from recent models like ex2 Lee et al. (2021), mT5 Xue et al.
(2020) and is most similar to the translate and fill approach suggested by
Nicosia et al. (2021). In our work instead of fine tuning the LM to generate
translations, we fine tune it to generate CS utterances.

###  2.2 Conversational Semantic Parsing

Today most commercial Lialin et al. (2020) conversational semantic parsing
systems utilize hierarchical representations, such as the TOP representation
Gupta et al. (2018), which is typically modelled using sequence to sequence
task Rongali et al. (2020); Shaw et al. (2020). Our semantic parsing model is
most similar to Cole et al. (2021) where we use a T5 model as the base
semantic parser.

##### Code-Switched Semantic Parsing

Traditionally, CS has been explored in word-level language identification
Molina et al. (2019) and named entity recognition Aguilar et al. (2019) and
shallow parsing Sharma et al. (2016); Bhat et al. (2018). Recent works like
Duong et al. (2017) introduced a CS test set for semantic parsing, Samanta et
al. (2019) introduce a variational auto-encoder based generation technique,
some other works take inspiration from machine translation Tarunesh et al.
(2021) and adversarial networks Chandu and Black (2020). These works rely on
specialized architectures and only produce code switched sentences, whereas
our work simply relies on a LM finetuning and outputs both sentences and
semantic parses.

More recently, Einolghozati et al. (2021) released a Spanglish semantic
parsing dataset named CSTOP and Mehnaz et al. (2021) released a Hinglish
conversation summarization dataset named GupShup. Similar to the previous
works, we release a TOPv2 derived code-switched dataset TOP-Hinglish. We also
release the synthetic augmented dataset which is an order of magnitude larger
than any such previous dataset.

##  3 The Code Switched Dataset

###  3.1 Collecting Hinglish-TOP

Domain |  Pre-processed English Utterance (Input) |  Code-Switched Utterance (Output)
---|---|---
Navigation (14.5%) |  What’s the traffic like on [Long Island]1 going to [the Hamptons]2 [tonight]3 |  [Aaj raat]3 [Hamptons]2 jaate hue [Long Island]1 par traffic kaisa hoga
Weather (16.6%) |  Whats the current weather conditions in [Arizona]2 ? |  [Arizona]2 me current mausam ka haal kya hai ?
Reminder (14.1%) |  Remind [me]1 to [send out the company meeting notes]2 [at 5 pm today]3. |  [Muje]1 aaj [shaam 5 baje]3 [company meeting note ko send]2 karne ka remind karaye
Alarm (18.2%) |  Set me an alarm [every Thursday at 5AM]1 [until the 1st July]2 |  Muje [1 july tak]2 ke liye [har thursday ko subah 5 baje]1 ka alarm set kare
Music (10.8%) |  Can I listen to the 5th [song]1 on the [album]2 one more time ? |  kya mai [album]2 par 5th [song]1 ek bar fir sun sakta hoon ?
Timer (10.9%) |  I want you to change the [timer]1 to a [30 minutes]2 instead of [15 minutes]3. |  Mai chahta hu ki aap [timer]1 ko [15 minutes]3 ke bajaye [30 minutes]2 me badal den.
Event (6.9%) |  When does [cider tasting]1 season start in [ Woodinville, Washington]2 |  [Woodinville , Washington]1 me [cider tasting]1 season kab shuru hoga
Messaging (8.0%) |  Send a message to [Diana]1 and [Rich]2 that [I will be earlier than expected to the library]3. |  [Diana]1 aur [Rich]2 ko message bhejo ki [mai library time se pehle pahunch jaunga]3
Table 2:  Examples of CS utterances obtained from annotators for the 8 domains
in the TOPv2 dataset. The relative sizes of each domain is in parenthesis. The
CS utterance is aligned with the English utterance’s parse to identify the CS
utterance’s semantic parse (Section 3.1.2).

For the Hinglish-TOP data collection, we randomly sampled a set of 10,896
utterances from the TOPv2 dataset Chen et al. (2020). These utterances are
distributed across 8 domains (as present in the original TOPv2 dataset). We
used a skewed train, validation and test set distribution to focus on getting
a bigger test set. Following this we sampled around 2993, 1390 and 6513
utterances from the train, validation and test splits of the original TOPv2
dataset. We observed that there were overlaps across the three splits in the
original TOPv2 dataset which we also preserved as is. A domain-wise breakdown
of the annotated CS data and a few examples can be found in Table 2.

Our data collection process can be broken down into the steps described below.
For these steps we preserved the domain and split information for the sampled
utterances. Our approach is also detailed in Figure 1.

####  3.1.1 Pre-Processing

As a pre-processing step, the slots and intents present in the semantic parse
of the English utterance were marked within the utterance using special span
identifiers (span-ID), which aligns their position in the semantic tree. By
marking these spans we know what words in the sentence correspond to the leaf
arguments and ask the raters to mark the same in the code-switched query. Some
examples of pre-processed English utterances can be seen in Table 2, 2nd
column. The pre-processing step allows us to reconstruct the semantic parse
for the code-switched utterance, described next.

####  3.1.2 Code-switching and Alignment

Native Hindi-English speakers were chosen as human annotators for code-
switching the English utterances. We asked the the annotators to generate a
hinglish code switched utterance for the given sentence while preserving the
naturalness and semantic equivalence of the code-switched utterance. This task
was done by 3 annotators. A primary annotator whose task is to code-switch the
English utterance. Additionally, we had one more annotator who checked the
utterances for naturalness and semantic equivalence. We keep the utterance
when both the checker agreed on the naturalness and semantic equivalence. The
definition of naturalness and semantic equivalence is same as in Section 3.3.
Some examples of code-switched utterances after the data annotation task are
shown in Table 2. This step is shown as Step 1 in Figure 1.

##### Annotation Alignment

After obtaining human annotated code-switched utterances with the slot spans
($3^{rd}$ column Table 2), we transfer the slot names (e.g., $date\\_time$,
$alarm\\_name$ etc.) from the English parse to the code-switched utterance to
obtain human labeled data for code-switched semantic parsing. As we preserved
the slot spans (Table 2) as the part of the parsing task there is a 1:1
mapping between the english parse the code switched utterance. The example
below denotes this step, we map each date_time slot to a unique span id to get
a 1:1 alignment.

[ Set alarm [ for 4:30 am on Tuesday ]date_time and [ Thursday ]date_time of
next week ]create_alarm
---
Set alarm [ for 4:30 am on Tuesday ]1 and [ Thursday ]2 of next week

At the end of this step, we had collected 10k CS utterances, along with the
semantic parses derived using the alignment above. Details on our human
annotated Hinglish-TOP dataset are in Table 4.

English Utterance |  Code-switched Utterance |  Rating
---|---|---
What is the weather in Canada ? |  Canada me mausam kaisa hai ? |  Natural & Semantically similar
How long is left before my alarm goes off ? |  Mera alarm kab tak band rahega ? |  Natural but Not Semantically similar: English utterance is asking for time remaining, while CS utterance is asking for trigger time
Remind me the first Tuesday of every month that bulk trash pick up is the next morning . |  Muje every month first Tuesday ko remind karaye ki bulk trash pick up is agley subha hai . |  Not Natural but Semantically similar: Phrase "bulk trash pick up is agley subha hai" is un-natural
How long of a delay is traffic causing |  traffic kaisa delay causing hai |  Not Natural & Not Semantically similar
Table 3:  Examples of CS utterances generated by the XXL mT5 model, along with their human ratings. English Vocabulary size | 4857
---|---
Romanized Hindi Vocabulary size | 1931
Total utterances | 10,896
Avg. # of Hindi tokens per utterance | 4.36
Avg. # of English tokens per utterance | 3.82
Avg. # of CS points per utterance | 3.56
Table 4: Dataset statistics for the human annotated Hinglish-TOP dataset.

###  3.2 CS Data Generation using mT5

English Utterance |  Generated Code-Switched Utterance |  Error Type
---|---|---
[ 9 pm ]1 [ appointment for photos ]2 and remind [ me ]3 [ an hour before ]4 |  [ mujhe ]3 [ 9 pm ]1 ko [ photos ke liye appointment ]2 hai aur [ mujhe ]3 [ ek ghante pehle ]4 yaad dilaayen |  Unequal number of argument spans in English (one [ me ]3) and code-switched query (two [ mujhe ]3)
play [ song ]1 [ Heart is on fire ]2 on [ spotify ]3 |  [ spotify ]3 par [ song ]1 [ Heart is on fire ]two ko bajao |  Improper span ID formatting for Code-Switched utterances
Change [ banking ]1 reminders [ from ]2 [ once a week ]3 [ to ]4 [ twice a week ]5 |  [ banking ]1 reminders ko [ [ ek bar har week ]3 [ dohrayen ]4 |  Unequal number of opening and closing spans for Code-Switched utterances
Remind [ me ]1 to [ email ]2 [ Michelle ]3 [ on Tuesday ]4 [ about ]5 [ the recital ]6 |  [ Mujhe ]1 [ Tuesday ko ]7 [ Michelle ]3 ko [ email ]2 karne ke liye yaad dilaayen |  Non-matching span IDs in english and code-switched utterances
Table 5:  Examples of errors that were automatically filtered from the mT5 generated code switched utterances.  Data source | Naturalness | Semantic Equiv.
---|---|---
Human | 99.2% (0.72) | 97.7% (0.69)
mT5 Small | 86.1% (0.8) | 43.7% (0.88)
mT5 XXL | 98.1% (0.95) | 89.5% (0.73)
Table 6:  Naturalness and Semantic Equivalence Statistics from Data Quality
Task. The $\kappa$ values for rater agreement is listed in brackets.

In order to generate synthetic code-switched utterances, we fine-tuned a mT5
Xue et al. (2020) model for the task of code-switching the English utterances
using the training data obtained in the previous step. The input to the T5
model is pre-processed English utterance (second column Table 2) and the
output is the code-switched utterance containing slot spans (3rd column Table
2). The TOPv2 dataset originally had over 180K utterances, of which we
manually annotated only 10.8K (3K train, 1.3K validation and 6.5K test split)
utterances. The annotated train set was used to fine-tune the model. The
remaining utterances from TOPv2 was used as monolingual input to the mT5 model
to generate synthetic code-switched utterances.

Since mT5 is a text to text model, the generated synthetic code-switched
utterances were not always structurally correct. This resulted in errors when
trying to align the output with the English semantic parse as mentioned in
Section 3.1.2.

####  3.2.1 Data Filtering

Seed Set Size |  Small mT5 Throughput |  XXL mT5 Throughput
---|---|---
100 |  47.3% |  82.0%
500 |  64.4% |  93.7%
1k |  72.8% |  96.1%
2k |  89.7% |  97.5%
3k |  92.7% |  98.3%
Table 7: Percentage of input English queries that get code switched post data-
filtering with varying number of seed examples for both XXL and small model.
We use the remaining TOPv2 dataset (173,042 utterances) for generating the CS
utterances.

Sometimes the code-switched utterances generated from the fine-tuned mt5 model
have structural errors which prevent aligning the semantic parse for these
code-switched utterances to their English counterparts. We filter out such
utterances using a syntactic rule based filter. In particular we removed
examples which contain:

  * •

Unequal number of argument spans in English and code-switched utterances.

  * •

Improper span-ID formatting (when the span-ID could not be extracted) for
code-switched utterances.

  * •

Unequal number of opening and closing spans for code-switched utterances.

  * •

Non-matching span-ID in English and code-switched utterances.

Examples of these errors can be seen in Table 5.

Finally, post data-filtering, we align the semantic parse annotations of the
English utterance to the code-switched utterances using the same approach as
in Section 3.1.2. The amount of syntactically correct synthetically generated
data different seed set sizes is shown in Table 7. Post data-filtering using
100 fine-tuning examples, the small mT5 model had a yield of 47.3% on the
remaining TOPv2 dataset, while the XXL model successfully converted 82.0% of
the English utterances.

###  3.3 Data Quality Assessment

We conduct both intrinsic and extrinsic evaluation of the generated synthetic
data. The results of extrinsic evaluation are presented in Section 5. In this
section we focus on 2 main questions:

What is the data quality of the human annotated data? To measure the quality
of the CS data we created a second annotation task in which speakers
proficient in both English and Hindi rated the CS utterances for their
naturalness and semantic equivalence to the original English utterance. For
naturalness, raters were just shown the CS utterance and were asked to answer
the question: Would a native speaker utter this utterance naturally in a
conversation? For semantic equivalence, raters were shown both the English and
the CS utterance and were asked: Do these sentences convey exactly the same
meaning? This serves as an upper bound on naturalness and semantic
equivalence. Example ratings for naturalness and semantic equivalence are
shown in Table 3.

We did this study on a sample of 2021 utterances. Each CS utterance was rated
by two human raters, and conflicts in opinion were resolved by using a third
rater and taking the majority opinion. For the human annotated data (Table 6,
the raters found over 99% of the sampled utterances to be natural as well as
semantically equivalent to the source English utterance.

How good are the synthetic CS utterances when judged by a human? We also
performed the same quality measurements on the mT5 generated data to estimate
naturalness and semantic equivalence. We randomly sampled 500 generated
queries from the small and XXL model trained using the largest seed set. We
observed that the data generated by the XXL model was of much higher quality
compared to the small model. For the XXL model, the raters found over 98% of
the sampled generated utterances to be natural and over 89% were semantically
equivalent to the source English utterance. Additionally, we observed high
$\kappa$ ($>$0.65) values for inter-annotator agreement; details can be found
in Table 6.

##  4 Experimental Setup

Additionally for extrinsic evaluation, we answer the following research
questions through our experiments: (a) What effect does adding augmented data
to the training set have on the semantic parsing task? (b) What is the effect
of varying the size of initial seed set for the data augmentation step on the
overall performance?; (c) How does the performance vary across domains?

To determine the quality of the augmented data, we trained semantic parsing
models for code-switched utterances. We fine tuned a second mT5 model for this
task. We compared the performance of our semantic parsing models with and
without augmented data while keeping everything else the same. To study the
efficacy of our technique with regards to the seed set, we trained models by
varying the seed set size 100, 500, 1000, 2000 and 3000 utterances. Further,
these training data of varying sizes were used to augment data using the un-
annotated utterances from the TOPv2 dataset in accordance to section 3.2.
Although we only release the augmented data from the model trained on full
3000 training examples, an independent augmented data set was created for each
batch of training data from the training split.

For the mT5 models, we use the public checkpoints provided by Xue et al.
(2020). We only roughly tuned the hparams. For the XXL (13 billion parameters)
mT5 model training, we use a learning rate of 0.001, batch size of 512 and
fine tune for 20k steps. For the small mT5 model (300 million parameters), we
fine tune for 200k steps keeping rest of the parameters exactly the same. We
checked for overfitting by choosing earlier (10k, 50k steps) checkpoints but
obtained best results with the parameters mentioned.

Human Annotated Utterances
---
EN: Will it be [ above ] [ 59 ] in [ celsius ] at [ Menlo Park ] [ tomorrow morning ] |  CS: Kya [ kal subah ] [ Menlo Park ] me [ 59 ] [ celsius ] se [ upar ] hoga?
EN: Remind [ me ] [ on September 1 ] that [ its my best friend’s birthday ] ! |  CS: [ September 1 ko ] [ muje ] remind karaye ki [ mere best friend ka birthday he ]
Model Generated Utterances
EN: I need to change [ my ] [ car maintenance ] reminder to [ Tuesday 9 AM ] |  CS: Mujhe [ mere ] [ car maintenance ] reminder ko [ Tuesday subah 9 bajhe ke liye ] badalne ki zaroorat hai
EN: Text [ my ] [ mom ] that [ I will um reach home in ten minutes ] |  CS: [ Meri ] [ ma ] ko text karo ki [ mai um ten minutes me pahuch jaunga ]
Table 8:  Example of Human annotated CS utterances and Model Generated CS
utterances, along with the original English utterance. Same slots have been
marked with the same colour across utterances.

##### Models used for comparison

For varying the size of the seed set, we trained three mT5 models.

  1. 1.

A CS utterance generation model, used to generate the augmented data for each
seed set.

  2. 2.

A semantic parser trained solely on the seed set.

  3. 3.

A semantic parser trained on the seed set and the augmented data.

The models were tested over the test data set from the data annotation task of
6.5k utterances.111Note that we report all results on this code switched human
annotated testset. We used the exact-match accuracy to judge the performance
of these models.

##  5 Results

$100$$500$$1{,}000$$2{,}000$$3{,}000$$20$$40$$60$$80$20x# Human Annotated
Examples in Seed SetExact Match Acc.Small no augmentationSmall with CST5
augmentationXXL no augmentationXXL with CST5 augmentation Figure 3: Comparison
of Exact Match Accuracy for semantic parsing models trained with and without
CST5 augmentation, for varying seed set size. The dotted line shows the XXL
model reaches EM of 60% using CST5 augmentation with 100 examples, equivalent
to fine tuning a XXL model with 2k examples.
$100$$500$$1{,}000$$2{,}000$$3{,}000$$20$$40$$60$$80$$90$$100$# Training
ExamplesExact Match Acc.alarmeventmessagingmusicnavigationremindertimerweather
Figure 4: Exact Match Accuracy for different domains in the Hinglish-TOP
dataset, with varying amount of seed training examples used for data
augmentation.

As seen from Figure 3, we observe that increasing the size of the training
data, improves the performance of both models(with and without augmented data
set) significantly. While there is a consistent improvement on adding more
training data, the improvement itself seems to diminish with more seed set
examples. Empirically, we observe that the models which make use of the
augmented data reaches saturation, i.e. the value after which additional data
points don’t add to the overall performance of the model, quicker than the
models which do not make use of the augmented data. This shows that the
synthetic data generated by our technique is useful for the end task of
semantic parsing.

As seen from Figure 3, the performance of the mT5 model trained using the
augmented data was consistently better than that trained without it. The XXL
model trained using our technique with a seed set of only 100 examples
performs similar to the model trained with 2000 human annotated examples. This
is over a 40% absolute improvement with 100 examples. Even with 3000 human
annotated examples we see over 15% improvement in EM accuracy. This goes to
show the effectiveness our data augmentation technique, especially with low-
resource settings.

The XXL mT5 model was able to produce a significantly higher number of code-
switched utterances compared to the small mT5 model, which ultimately led to
the significant gain in its performance. The number of generated code-switched
utterances post data-filtering can be seen in Table 7. As seen from the
results (Figure 3) the XXL mT5 model is significantly better than the small
mT5 model, both as a parser and a data generator. We also observed that adding
the augmented data, allows the model to handle more complex queries. A few
examples from the models trained on full 3000 training examples have been
included in Table 8.

Moreover, after an in-depth domain-wise analysis of the models performance
(Figure 4), we observed that the model struggled with the reminder domain. It
is interesting to note that the reminder domain had the highest number of
unique intents, 16. Also, reminder domain had the highest number of average
intents per utterance of 2.66. This collectively led to the low performance of
the model on this domain. Some of the domains saturated quicker and did not
see much improvement upon adding more data points. We can empirically observe
that for domains with more inherent difficulty (reminder for example) the
value of added data is is more than simpler domains (weather for example).

##### Results on CSTOP

Additionally, we repeated a similar experiment for CSTOP Einolghozati et al.
(2021), which contains spanish-english code switched utterances annotated in
the TOP schema. Unlike our approach CSTOP doesn’t have parallel data with
TOPv2. Since only the weather domain was common between CSTOP and TOPV2, we
work with the weather domain subset. As we need parallel english data to
finetune the mT5 model, we used google translate to generate parallel english
data for 100 spanish-english queries. We used this to finetune a mT5 model and
generated new synthetic data using TOPv2 weather domain as the monolingual
dataset. For semantic parsing using 100 queries from the training data we
observe similar trends as Hinglish. We observe an improvement in the parsing
performance from 69.2% EM to 77.8 % EM on the testset using the synthetic code
switched data .

##  6 Conclusion

We proposed CST5, an approach to overcome the scarcity of labeled data for
code-switched semantic parsing. CST5 generates code switched utterances from
English utterances using a large pretrained LM mT5, and a small number of seed
examples. We showed that the generated utterances were of high quality, as
determined by human raters in a quality annotation task. By aligning the
generated utterances with the semantic parse of the original English
utterance, we derived a large supervised dataset for training a code-switched
semantic parser using a labeled dataset in English.

We did both intrinsic and extrinsic evaluation of the human annotated as well
as the synthetic data. Human raters found 98% of the synthetic data to be
natural and 89% to be semantically equivalent. Extrinsically, our experiments
also demonstrated that data augmentation using CST5 effectively reduces the
data requirements by 20x for the code switched semantic parsing task. To
further research in this area, we will release the dataset of over 10k
manually annotated Hinglish utterances, along with over 170k examples
generated using CST5222This data is available at https://github.com/google-
research-datasets/Hinglish-TOP-Dataset. While we applied CST5 to Hindi-English
code-switching, our approach is general and can be applied to any other
language pair that exhibits code-switching. In the spirit of CST5, we believe
that using large pre-trained language models to perform data augmentation for
other code-switched NLP tasks is an attractive future direction to explore.

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# Study of the Ultraviolet Behavior of an O($N$) $|\vec{\phi}|^{6}$ Theory in
$d=3$ Dimensions

Robert Shrock  C. N. Yang Institute for Theoretical Physics and Department of
Physics and Astronomy,
Stony Brook University, Stony Brook, NY 11794, USA

###### Abstract

We study the ultraviolet (UV) behavior of an O($N$) $|\vec{\phi}|^{6}$ theory
in $d=3$ spacetime dimensions, focusing on the question of the range in $N$
over which the perturbative beta function exhibits robust evidence of a UV
zero in the $|\vec{\phi}|^{6}$ coupling, $g$. The four-loop $(4\ell)$ beta
function is known to have a (scheme-independent) UV zero at
$g=g_{{}_{UV,4\ell}}$, which is reliably calculable for large $N$. For our
analysis we use the six-loop beta function calculated in the minimal
subtraction scheme. We find that this six-loop beta function has a UV zero,
$g_{{}_{UV,6\ell}}$, if $N>N_{c}$, where $N_{c}\simeq 796$, and we calculate
$g_{{}_{UV,6\ell}}$. To investigate the reliability of the result in the
region of
$N\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}N_{c}$,
we apply three methods: (i) calculation of the fractional difference between
$g_{{}_{UV,4\ell}}$ and $g_{{}_{UV,6\ell}}$, (ii) a Padé approximant, and
(iii) an assessment of scheme dependence. Our results provide quantitative
measures of the range of $N$ over which the six-loop beta function has a UV
zero and of the $1/N$ corrections to the value of $g$ at the UV zero for large
but finite $N$. If one imposes a benchmark requirement that the fractional
difference between $g_{{}_{UV,4\ell}}$ and $g_{{}_{UV,6\ell}}$ must be less
than 15 %, then our results show that this requirement is satisfied for
$N\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}2\times
10^{3}$. The possible role of nonperturbative effects is also noted.

##  I Introduction

In this paper we study the ultraviolet (UV) behavior of an O($N$)
$|\vec{\phi}|^{6}$ quantum field theory in $d=3$ spacetime dimensions. This
theory, commonly denoted $|\vec{\phi}|^{6}_{3}$, involves an $N$-component
real scalar field $\vec{\phi}=(\phi_{1},...,\phi_{N})^{T}$ and is defined by
the path integral $Z=\int\prod_{x}[d\phi_{i}(x)]\,e^{iS}$ with $S=\int
d^{3}x\,{\cal L}$, and the bare Lagrangian

${\cal
L}=\frac{1}{2}(\partial_{\nu}{\vec{\phi}})\cdot(\partial^{\nu}{\vec{\phi}})-\frac{1}{2}m^{2}|\vec{\phi}|^{2}-\frac{\lambda}{4N}|\vec{\phi}|^{4}-\frac{g}{6N^{2}}|\vec{\phi}|^{6}\
,$ (1)

where $|\vec{\phi}|=(\sum_{i=1}^{N}\phi_{i}^{2}\,)^{1/2}$. In $d=3-\epsilon$
(Euclidean) dimensions, the O($N$) $|\vec{\phi}|^{6}$ theory has been
extensively analyzed to obtain expressions for critical exponents describing
tricritical points in condensed matter physics Stephen and McCauley (1973);
Lewis and Adams (1978); Lawrie and Sarbach (1984); Hager (2002). Early studies
of the theory as a relativistic quantum field theory in $d=3$ spacetime
dimensions include Wilson (1972); Townsend (1977); Pisarski (1982); Appelquist
and Heinz (1981, 1982); Bardeen  _et al._ (1984).

Because of quantum corrections, the physical coupling $g=g(\mu)$ depends on
the Euclidean energy/momentum scale, $\mu$, where it is measured. This
dependence is described by the renormalization group (RG) beta function Callan
(1970); Symanzik (1970); Wilson (1971) of the theory, $\beta_{g}=dg/d\ln\mu$.
The lowest-order (two-loop, $O(g^{2})$) term in $\beta_{g}$ is positive
Stephen and McCauley (1973), so this theory is infrared (IR)-free, i.e.,
$g(\mu)\to 0$ as $\mu\to 0$. An important question is whether, for a given
$N$, the theory has a UV zero in $\beta_{g}$ at some value $g_{{}_{UV}}$. If
this is the case and if a perturbative analysis is adequate to describe the
physics, then, as the reference energy/momentum scale $\mu$ increases from 0
to $\infty$, $g(\mu)$ increases from 0 and approaches $g_{{}_{UV}}$ from
below. Since the coefficients of the quadratic and quartic terms in the
Lagrangian (1) are both dimensionful, and since
$\lim_{\mu\to\infty}m^{2}/\mu^{2}=0$ and $\lim_{\mu\to\infty}\lambda/\mu=0$,
they are expected to play a negligible role in the ultraviolet limit
$\mu\to\infty$. We denote the UV zero (presuming that it exists) of the
$n$-loop ($n\ell$) beta function as $g_{{}_{UV,n\ell}}$. The term of order
$g^{p}$ in $\beta_{g}$ arises from graphs with a maximum number of loops $n$
given by $n=2(p-1)$. The $O(g^{3})$ term in the beta function is negative, so
that at this order, this four-loop ($4\ell$) beta function, $\beta_{g,4\ell}$,
has a UV zero Townsend (1977); Pisarski (1982); Appelquist and Heinz (1982).
In the large-$N$ limit, with the normalization in Eq. (1), this occurs at the
value of the coupling $g_{{}_{UV,4\ell}}=192$. It was noted in Townsend
(1977); Pisarski (1982) that the $N$-dependence of higher-loop terms in
$\beta_{g}$ is such that for large $N$, the inclusion of these higher-loop
terms would produce only a small fractional shift $\propto 1/N$ in the value
of the coupling at the UV zero, and therefore the calculation of the value
should be reliable in the large-$N$ limit. Such a UV zero in the beta function
is a UV fixed point (UVFP) of the renormalization group. The existence of a
UVFP in an IR-free theory is of considerable interest, since it means that one
has perturbative control of the theory in both the IR and UV limits. A
previous example of an IR-free theory with a UVFP is the nonlinear $O(N)$
$\sigma$ model in $d=2+\epsilon$ dimensions Brézin and Zinn-Justin (1976a, b);
Bardeen  _et al._ (1976); Polyakov (1975). The early studies Townsend (1977);
Pisarski (1982) also cautioned that at small and moderate $N$, this formal UV
zero at $g=g_{{}_{UV,4\ell}}$ might be an artifact of the perturbative
calculation.

We briefly review some further relevant work on this theory. After the studies
in Townsend (1977); Pisarski (1982); Appelquist and Heinz (1981, 1982), a
variational calculation was carried out in the $N\to\infty$ limit by Bardeen,
Moshe, and Bander (BMB) in Bardeen  _et al._ (1984), who found that for
$g>g_{cr}$, where $g_{cr}=(4\pi)^{2}\simeq 158$, the theory undergoes a
transition to a strongly coupled phase involving dynamical mass generation for
the scalar field and spontaneous breaking of scale invariance, with the
resultant appearance of a massless Nambu-Goldstone boson (NGB), namely a
dilaton Bardeen  _et al._ (1984). Since $g_{cr}<g_{{}_{UV,4\ell}}$, it was
concluded in Bardeen  _et al._ (1984) that in the large-$N$ limit where the
BMB calculations were performed, the physics is described by the properties of
this strong-coupling phase rather than by a UV zero in the (perturbative) beta
function. The properties in the $N\to\infty$ limit were further studied in
David  _et al._ (1984, 1985); Amit and Rabinovici (1985). Exploratory lattice
studies to probe the BMB phase were performed in Kessler and Neuberger (1985);
Karsch and Meyer-Ortmanns (1987). Refs. Gudmundsdottir  _et al._ (1984, 1985)
argued that at finite $N$, the BMB phase is unstable. More recently, Ref. Omid
_et al._ (2016) investigated the effect of higher-order corrections in $1/N$
on the BMB dilaton and found that it becomes a tachyon when one takes account
of these $1/N$ corrections. On this basis, the authors of Ref. Omid  _et al._
(2016) concluded that at finite $N$, the BMB phase with spontaneously broken
approximate scale invariance is unstable. Some recent related studies of this
theory include Litim  _et al._ (2017); Litim and Trott (2018); Yabunaka and
Delamotte (2017, 2018); Fleming  _et al._ (2020).

In parallel with these continuing studies of the role of possible
nonperturbative effects at finite $N$ in the $|\vec{\phi}|^{6}_{3}$ theory, it
seems worthwhile to investigate the UV properties of the (perturbative) beta
function further. At the time of the early studies in Townsend (1977);
Pisarski (1982); Appelquist and Heinz (1981, 1982); Bardeen  _et al._ (1984),
$\beta_{g}$ had been calculated only up to $O(g^{3})$. Subsequently, it was
calculated to $O(g^{4})$ in Hager (2002), and this remains the highest order
to which it has been computed. A very basic question that, to our knowledge,
has not been studied yet, is whether, for a given $N$, this $O(g^{4})$ beta
function exhibits evidence for a (reliably calculable) UV zero, denoted
$g_{{}_{UV,6\ell}}$. We address this question in this paper.

A necessary condition for such evidence is that the values of the coupling at
this UV zero obtained from calculations of the beta function to successive
orders in $g$ should be close to each other. To determine the region in $N$
over which this condition is satisfied, we will compare the values of
$g_{{}_{UV,4\ell}}$ and $g_{{}_{UV,6\ell}}$ (for $N$ values where the latter
exists). Furthermore, the terms in the beta function at order $g^{p}$ with
$p\geq 4$ depend on the scheme used for regularization and renormalization. By
itself, this property does not render these higher-order terms unphysical; for
example, higher-order calculations of quark and gluon scattering in quantum
chromodynamics (QCD) are also scheme-dependent, but still play a crucial role
in the analysis of experimental data, and work continues on the construction
and application of optimal schemes for QCD calculations (see, e.g., Brodsky
_et al._ (2014) and references therein). However, this does mean that one must
assess the effect of this scheme-dependence, and we shall do this as part of
our study. In carrying out this study, it should be stressed that
nonperturbative effects may be important, and we refer the reader to the
continuing analysis of this topic, e.g. in Omid  _et al._ (2016); Fleming  _et
al._ (2020), as well as earlier works including Bardeen  _et al._ (1984).
However, bearing this caveat in mind, one should at least elucidate the
predictions from the beta function calculated to the highest order to which it
is known for general $N$, and that is the purpose of our present study.

In passing, it should be mentioned that theories of $\phi^{6}_{3}$ type with
various global symmetries, representations of the scalar fields, and sixth-
degree interaction terms have also been of recent interest in the context of
large-charge expansions and conformal field theory, e.g., Hellerman  _et al._
(2015); Alvarez-Gaumé  _et al._ (2017); Badel  _et al._ (2020); Antipin  _et
al._ (2020); Jack and Jones (2020). Here we will confine our analysis to the
simple realization of this theory in the Lagrangian (1).

This paper is organized as follows. In Section II we present the results of an
analysis of the evidence, for a given $N$, of a UV zero the six-loop beta
function. In Section III we apply the method of Padé approximants to study
this question further. Section IV contains an assessment of the effects of
scheme dependence. Our conclusions are presented in Section V. Some auxiliary
relations are given in an Appendix.

##  II Beta Function and UV Zero

In this section we analyze the beta function of the $O(N)$
$|\vec{\phi}|^{6}_{3}$ theory. The beta function $\beta_{g}=dg/d\ln\mu$ has a
series expansion in powers of the interaction coupling $g$, starting with a
term of $O(g^{2})$,

$\beta_{g}=g\sum_{j=1}^{\infty}b_{j}g^{j}\ .$ (2)

In the literature there are several different normalization conventions for
the $|\vec{\phi}|^{6}$ coupling; for the reader’s convenience, in the Appendix
we list conversion formulas relating some of these. As noted above, the term
in $\beta_{g}$ of $O(g^{p})$ arises from graphs with a maximal number of loops
equal to $n=2(p-1)$. We denote the truncation of the infinite series (2) to
$O(g^{p})$, as $\beta_{g,n\ell}$, where $n\ell$ is short for $n$-loops. As is
the case with other scalar field theories, although Eq. (2) is an asymptotic
expansion Lipatov (1977); Brézin  _et al._ (1977), it can still yield useful
information about the properties of the theory.

With the normalization in Eq. (1), the first two coefficients, $b_{j}$,
$j=1,2$, are Townsend (1977); Pisarski (1982); Appelquist and Heinz (1982)
(see also Stephen and McCauley (1973); Lewis and Adams (1978))

$b_{1}=\frac{3N+22}{2\pi^{2}N^{2}}$ (3)

and

$\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2^{7}\pi^{2}N^{4}}\bigg{[}(N^{3}+34N^{2}+620N+2720)+\frac{8}{\pi^{2}}(53N^{2}+858N+3304)\bigg{]}$
(4) $\displaystyle=$ $\displaystyle\frac{1}{N^{4}}\bigg{[}(0.791572\times
10^{-3})N^{3}+0.0609195N^{2}+1.04129N+4.27300\bigg{]}\ ,$ (6)

where floating-point numbers are given to the indicated accuracy. As mentioned
before, these first two terms of $O(g^{2})$ and $O(g^{3})$ in $\beta_{g}$
contain the maximal scheme-independent information in this function. The UV
zero in the four-loop beta function occurs at
$g_{{}_{UV,4\ell}}=-b_{1}/b_{2}$, namely

$g_{{}_{UV,4\ell}}=\frac{64N^{2}(3N+22)}{(N^{3}+34N^{2}+620N+2720)+(8/\pi^{2})(53N^{2}+858N+3304)}\
.$ (7)

This is a monotonically decreasing function of $N$. For large $N$,

$\displaystyle g_{{}_{UV,4\ell}}$ $\displaystyle=$ $\displaystyle
192\bigg{[}1-\frac{8(159+10\pi^{2})}{3\pi^{2}N}+\frac{4(134832+14144\pi^{2}+215\pi^{4})}{3\pi^{4}N}+O\Big{(}\frac{1}{N^{3}}\Big{)}\bigg{]}$
(8) $\displaystyle=$ $\displaystyle
192\bigg{[}1-\frac{69.62685}{N}+\frac{4043.0263}{N^{2}}+O\Big{(}\frac{1}{N^{3}}\Big{)}\bigg{]}\
.$ (10)

The coefficient $b_{3}$ of the $g^{4}$ term in $\beta_{g}$ has been calculated
in Hager (2002) in the minimal subtraction scheme ’t Hooft and Veltman (1972);
Bardeen  _et al._ (1978). With the normalization in Eq. (1), it is

$\displaystyle b_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{7}\pi^{6}N^{6}}\Big{(}1857N^{3}+45976N^{2}+367716N+950576\Big{)}+\frac{1}{2^{9}\pi^{4}N^{6}}\Big{(}36N^{4}+1607N^{3}+33568N^{2}+273772N+735392\Big{)}$
(11) $\displaystyle-$
$\displaystyle\frac{3\ln(2)}{2^{8}\pi^{4}N^{6}}\Big{(}N^{4}+N^{3}-700N^{2}-8236N-24816\Big{)}+\frac{5}{2^{11}\pi^{2}N^{6}}\Big{(}N^{4}+64N^{3}+1352N^{2}+12248N+36960\Big{)}$
(13) $\displaystyle-$
$\displaystyle\frac{105\zeta(3)}{2^{8}\pi^{6}N^{6}}\Big{(}11N^{3}+428N^{2}+4228N+12208\Big{)}+\frac{1}{2^{6}\pi^{6}N^{6}}\Big{(}24\beta(4)+\pi^{2}G\Big{)}\Big{(}31N^{3}+1126N^{2}+11876N+37592\Big{)}\
,$ (15) $\displaystyle=$ $\displaystyle\frac{1}{N^{6}}\Big{[}(0.885804\times
10^{-3})N^{4}+0.0739318N^{3}+1.81979N^{2}+16.3518N+47.4455\,\Big{]}\ ,$ (17)

where the Dirichlet beta function $\beta(s)$ is defined as

$\beta(s)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}}\ ,$ (18)

and $G=\beta(2)$ is the Catalan constant, with the values $G=0.915965594$ and
$\beta(4)=0.98894455$ to the given floating-point accuracy. As is evident from
the numerical evaluation of $b_{3}$ in Eq. (17), it is positive for all
physical $N$ and is a monotonically increasing function of $N$. With the
normalization convention in Eq. (1), these coefficients have the large-$N$
behavior $b_{1}\sim 1/N$, $b_{2}\sim 1/N$, and $b_{3}\sim 1/N^{2}$.

We now address the question of whether, for a given $N$, the six-loop beta
function, $\beta_{g,6\ell}$, with $b_{3}$ computed in the minimal subtraction
scheme, has a (perturbatively reliably calculable) UV zero. Aside from the
double IR-zero at $g=0$, the zeros of the $n$-loop beta function
$\beta_{g,n\ell}$, i.e., the beta function calculated to $O(g^{p})$, where
$p=(n/2)+1$, are the solutions of the algebraic equation
$\sum_{j=1}^{n/2}b_{j}g^{j-1}=0$. For $\beta_{g,6\ell}$, this is the equation
$b_{1}+b_{2}g+b_{3}g^{2}=0$, with solutions
$g_{\pm}=(2b_{3})^{-1}(-b_{2}\pm\sqrt{b_{2}^{2}-4b_{1}b_{3}}\,)$. With $N$
formally generalized from positive integers to positive real numbers,
$b_{2}^{2}-4b_{1}b_{3}$ is positive if $N>N_{c}=796.111$ and negative for the
rest of the physical range $1\leq N<N_{c}$. Hence, for $N>N_{c}$, the six-loop
beta function $\beta_{g,6\ell}$ has two zeros on the positive $g$ axis, and
the one nearer to the origin is the UV zero,
$g_{{}_{UV,6\ell}}=g_{-}=(2b_{3})^{-1}(-b_{2}-\sqrt{b_{2}^{2}-4b_{1}b_{3}}\,)$.
This has the large-$N$ series expansion

$\displaystyle g_{{}_{UV,6\ell}}$ $\displaystyle=$ $\displaystyle
192\bigg{[}1+\frac{4(978+25\pi^{2}-216\ln
2)}{3\pi^{2}N}+\frac{4}{3\pi^{4}N^{2}}\Big{(}3232704+212612\pi^{2}+4565\pi^{4}+8928\pi^{2}G-1218249\ln
2$ (19) $\displaystyle+$ $\displaystyle 124416(\ln 2)^{2}-33192\pi^{2}\ln
2-83160\zeta(3)\Big{)}+O\Big{(}\frac{1}{N^{3}}\Big{)}\bigg{]}$ (21)
$\displaystyle=$ $\displaystyle
192\bigg{[}1+\frac{145.230}{N}+\frac{67847.343}{N^{2}}+O\Big{(}\frac{1}{N^{3}}\Big{)}\bigg{]}\
.$ (23)

It follows that

$\lim_{N\to\infty}g_{{}_{UV,6\ell}}=\lim_{N\to\infty}g_{{}_{UV,4\ell}}=192\ .$
(24)

Note that the $1/N$ correction to $g_{{}_{UV,4\ell}}$ is negative, while the
$1/N$ correction to $g_{{}_{UV,6\ell}}$ is positive. Given the $N$-dependence
of still higher-order terms $O(g^{p})$ with $p\geq 5$ in $\beta_{g}$, as
discussed in Pisarski (1982), the result (24) can be generalized to

$\lim_{N\to\infty}g_{{}_{UV,n\ell}}=\lim_{N\to\infty}g_{{}_{UV,4\ell}}=192\ ,$
(25)

where $n=2(p-1)$ with $p\geq 4$.

A necessary condition for a credible zero of a beta function is that when one
calculates it to successive orders of perturbation theory, one obtains values
that are close to each other, i.e., values with small fractional differences.
We define the fractional difference as

$\Delta(g_{{}_{UV,n\ell}},g_{{}_{UV,n^{\prime}\ell}})\equiv\frac{g_{{}_{UV,n^{\prime}\ell}}-g_{{}_{UV,n\ell}}}{g_{{}_{UV,n\ell}}}\
.$ (26)

In the large-$N$ limit, the fractional difference between $g_{{}_{UV,4\ell}}$
and $g_{{}_{UV,6\ell}}$ is

$\displaystyle\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,6\ell}})$ $\displaystyle=$
$\displaystyle\frac{12(144+5\pi^{2}-24\ln
2)}{\pi^{2}N}+\frac{4}{\pi^{4}N^{2}}\Big{(}1215792+84036\pi^{2}+1850\pi^{4}+2976\pi^{2}G-436608\ln
2$ (27) $\displaystyle+$ $\displaystyle 41472(\ln 2)^{2}-12984\pi^{2}\ln
2-27729\zeta(3)+71424\beta(4)\Big{)}+O\Big{(}\frac{1}{N^{3}}\Big{)}$ (29)
$\displaystyle=$
$\displaystyle\frac{214.8566}{N}+\frac{78764.106}{N^{2}}+O\Big{(}\frac{1}{N^{3}}\Big{)}\
.$ (31)

This fractional difference vanishes as $N\to\infty$, in agreement with the
conclusion reached in Townsend (1977); Pisarski (1982); Appelquist and Heinz
(1982) (before $b_{3}$ had been calculated). The new information obtained here
is the calculation of the series expansion in powers of $1/N$ in Eq. (31),
which provides a quantitative measure of the accuracy and reliability of the
perturbative calculation of the value of the coupling at the UV zero for a
given large $N$. In Table 1 we list $g_{{}_{UV,4\ell}}$, $g_{{}_{UV,6\ell}}$,
and the fractional difference $\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,6\ell}})$
for some illustrative values of $N$. As $N$ increases well beyond $N_{c}$,
this fractional difference decreases reasonably quickly. For example, for
$N=2\times 10^{3}$, $N=4\times 10^{3}$, and $N=10^{4}$,
$\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,6\ell}})$ has the approximate values 13 %,
6 %, and 2 %, respectively. Thus, if one imposes a requirement that the
fractional difference $\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,6\ell}})$ must be
less than, say, 15 %, in order for the calculation of the value of the UV zero
to be reasonably reliable, then our results show that this criterion is
satisfied for
$N\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}2\times
10^{3}$.

##  III Analysis With Padé Approximants

One can gain further insight into the behavior of the beta function by the use
of Padé approximants. Given a series expansion $f(z)=\sum_{n=0}^{n_{\rm
max}}a_{n}z^{n}$, the $[r,s]$ Padé approximant (PA) is the rational function
with numerator and denominator polynomials in $z$ of degree $r$ and $s$,
respectively, where $r+s=n_{\rm max}$, such that the Taylor series expansion
of this rational function matches the series expansion for $f(z)$ to its
highest order, $n_{\rm max}$. The Padé method can be considered to be semi-
perturbative, since it uses as input a perturbative series expansion but
produces a closed-form rational function, whose higher-order terms of order
$z^{n}$ with $n>n_{\rm max}$ are thus determined. Since the double IR zero in
$\beta_{g}$ at $g=0$ is not relevant here, it will be convenient to consider
the reduced (red.) beta function normalized so that it is equal to 1 for
$g=0$:

$\beta_{g,{\rm
red.}}\equiv\frac{\beta_{g}}{\beta_{g,2\ell}}=\frac{\beta_{g}}{b_{1}g^{2}}=1+\frac{1}{b_{1}}\sum_{j=2}^{\infty}b_{j}g^{j-1}\
.$ (32)

From the beta function calculated to $O(g^{p})$, one thus obtains the reduced
beta function of degree $(p-2)$ in $g$. In particular, from $\beta_{g,6\ell}$,
we have $\beta_{g,{\rm red.},6\ell}=1+(b_{2}/b1)g+(b_{3}/b_{1})g^{2}$. We
denote the Padé approximants to $\beta_{g,{\rm red.},n\ell}$ simply as
$[r,s]_{n\ell}$. The PA $[2,0]_{6\ell}$ is this function itself, which we have
already analyzed, and the PA $[0,2]_{6\ell}$ has no zero, so we study the
$[1,1]_{6\ell}$ Padé approximant. In terms of the coefficients $b_{j}$,
$j=1,2,3$, this is

$[1,1]_{6\ell}=\frac{1+\Big{(}\frac{b_{2}^{2}-b_{1}b_{3}}{b_{1}b_{2}}\Big{)}g}{1-\Big{(}\frac{b_{3}}{b_{2}}\Big{)}g}\
.$ (33)

We label the zero of this $[1,1]_{6\ell}$ PA as $g_{{}_{UV,[1,1]_{6\ell}}}$.
This is

$g_{{}_{UV,[1,1]_{6\ell}}}=\frac{b_{1}b_{2}}{b_{1}b_{3}-b_{2}^{2}}=\frac{-\Big{(}\frac{b_{1}}{b_{2}}\Big{)}}{1-\Big{(}\frac{b_{1}b_{3}}{b_{2}^{2}}\Big{)}}\
.$ (34)

We list illustrative values of $g_{{}_{UV,[1,1]_{6\ell}}}$ in Table 1. As is
evident from the last term in Eq. (34), $g_{{}_{UV,[1,1]_{6\ell}}}$ is related
to the value of the UV zero of the four-loop beta function,
$g_{{}_{UV,4\ell}}=-b_{1}/b_{2}$, via division by the factor
$1-(b_{1}b_{3}/b_{2}^{2})$. Now $(b_{1}b_{3}/b_{2}^{2})>0$, so if
$(b_{1}b_{3}/b_{2}^{2})<1$, then
$g_{{}_{UV,[1,1]_{6\ell}}}>g_{{}_{UV,4\ell}}$. Since $b_{1}b_{3}/b_{2}^{2}\sim
O(N^{-1})$ as $N\to\infty$, it follows that

$\lim_{N\to\infty}g_{{}_{UV,[1,1]_{6\ell}}}=\lim_{N\to\infty}g_{{}_{UV,4\ell}}\
.$ (35)

For $N\gg 1$, $g_{{}_{UV,[1,1]_{6\ell}}}$ has the expansion

$\displaystyle g_{{}_{UV,[1,1]_{6\ell}}}$ $\displaystyle=$ $\displaystyle
192\Big{[}1+\frac{4(978+25\pi^{2}-216\ln
2)}{3\pi^{2}N}+\frac{4}{3\pi^{4}N^{2}}\Big{(}993216+57092\pi^{2}+1865\pi^{4}+8928\pi^{2}G-471744\ln
2$ (36) $\displaystyle+$ $\displaystyle 62208(\ln
2)^{2}-83160\zeta(3)+214272\beta(4)\,\Big{)}+O\Big{(}\frac{1}{N^{3}}\Big{)}\Big{]}$
(38) $\displaystyle=$ $\displaystyle
192\Big{[}1+\frac{145.230}{N}+\frac{21683.974}{N^{2}}+O\Big{(}\frac{1}{N^{3}}\Big{)}\Big{]}\
.$ (40)

As is evident from Eqs. (23) and (40), $g_{{}_{UV,6\ell}}$ and
$g_{{}_{UV,[1,1]_{6\ell}}}$ have the same leading $1/N$ correction terms. This
can be understood as a consequence of the fact that the $[1,1]_{6\ell}$ Padé
approximant incorporates information from the $b_{3}g^{4}$ term in
$\beta_{g,6\ell}$, or equivalently, the $(b_{3}/b_{1})g^{2}$ term in
$\beta_{g,{\rm red.},6\ell}$.

For large $N$, the fractional differences
$\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,[1,1]_{6\ell}}})$ and
$\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,[1,1]_{6\ell}}})$ are

$\displaystyle\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,[1,1]_{6\ell}}})$
$\displaystyle=$ $\displaystyle\frac{12(144+5\pi^{2}-24\ln
2)}{\pi^{2}N}+\frac{4}{\pi^{4}N^{2}}\Big{(}469296+32196\pi^{2}+950\pi^{4}+2976\pi^{2}G-187776\ln
2$ (41) $\displaystyle+$ $\displaystyle 20736(\ln 2)^{2}-4344\pi^{2}\ln
2-27729\zeta(3)+71424\beta(4)\Big{)}+O\Big{(}\frac{1}{N^{3}}\Big{)}$ (43)
$\displaystyle=$
$\displaystyle\frac{214.857}{N}+\frac{32600.74}{N^{2}}+O\Big{(}\frac{1}{N^{3}}\Big{)}$
(45)

and

$\displaystyle\Delta(g_{{}_{UV,6\ell}},g_{{}_{UV,[1,1]_{6\ell}}})$
$\displaystyle=$ $\displaystyle-\frac{144(144+5\pi^{2}-24\ln
2)^{2}}{\pi^{4}N^{2}}+O\Big{(}\frac{1}{N^{3}}\Big{)}$ (46) $\displaystyle=$
$\displaystyle-\frac{46163.369}{N^{2}}+O\Big{(}\frac{1}{N^{3}}\Big{)}\ .$ (48)

Since $g_{{}_{UV,6\ell}}$ and $g_{{}_{UV,[1,1]_{6\ell}}}$ have the same $1/N$
correction terms in the large-$N$ limit, as observed above, it follows that
$\Delta(g_{{}_{UV,6\ell}},g_{{}_{UV,[1,1]_{6\ell}}})$ vanishes like $1/N^{2}$
in this limit. This is in contrast to
$\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,{6\ell}}})$ and
$\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,[1,1]_{6\ell}}})$, which both vanish like
$1/N$ for large $N$.

In order for $g_{{}_{UV,[1,1]_{6\ell}}}$ to be acceptable as a UV zero, it is
necessary that there should not be a pole in the $[1,1]_{6\ell}$ PA on the
positive real axis closer to the origin. The pole in this approximant occurs
at

$g_{[1,1]_{6\ell},{\rm pole}}=\frac{b_{2}}{b_{3}}\ .$ (49)

Since $b_{2}<0$ and $b_{3}>0$ for all physical $N$, this pole occurs on the
negative real axis, thereby fulfilling the above necessary condition. As
$N\to\infty$, the value of $g$ at this pole behaves as

$\displaystyle g_{[1,1]_{6\ell},{\rm pole}}$ $\displaystyle=$
$\displaystyle-\frac{16\pi^{2}N}{144+5\pi^{2}-24\ln 2}+O(1)$ (50)
$\displaystyle=$ $\displaystyle-0.8936N+O(1)\ .$ (52)

In general, in the large-$N$ limit, the $1/N$ expansions given above show that

$g_{{}_{UV,4\ell}}\leq g_{{}_{UV,[1,1]_{6\ell}}}\leq g_{{}_{UV,6\ell}}\ .$
(53)

and

$|\Delta(g_{{}_{UV,6\ell}},g_{{}_{UV,[1,1]_{6\ell}}})|\leq\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,[1,1]_{6\ell}}})\
.$ (54)

with equality at $N=\infty$.

As $N$ decreases from large values, $b_{1}b_{3}/b_{2}^{2}$ increases, and as
$N$ decreases below a value $N_{d}\simeq 150.799$, this ratio increases
through 1, producing a pole in $g_{{}_{UV,[1,1]_{6\ell}}}$. Clearly, this
method of obtaining an estimate of a UV zero in $\beta_{g,6\ell}$ via a zero
in the $[1,1]_{6\ell}$ approximant at $g_{{}_{UV,[1,1]_{6\ell}}}$ is only
reliable for values of $N$ well above $N_{d}$.

Thus, the $[1,1]_{6\ell}$ Padé approximant to $\beta_{g,{\rm red.},6\ell}$
yields a UV zero over a larger range of $N$ than the beta function itself,
extending below $N_{c}\simeq 796$ to the vicinity of $N_{d}\simeq 151$.
However, as noted above, as $N$ approaches the vicinity of $N_{d}$ from above,
the value of $g_{{}_{UV,[1,1]_{6\ell}}}$ deviates substantially from the
scheme-independent value, $g_{{}_{UV,4\ell}}$. For example, at an illustrative
value below $N_{c}$ but above $N_{d}$, namely $N=300$, although the [1,1] Padé
approximant has a UV zero, $g_{{}_{UV,[1,1]_{6\ell}}}=378.17$, this is not
close to $g_{{}_{UV,4\ell}}=154.71$. Consequently, in this vicinity, the
method does not satisfy the requirement that different perturbative or semi-
perturbative methods of calculating this UV zero should yield values in
approximate agreement with each other. Among the entries in Table 1, in
addition to the values of $g_{{}_{UV,[1,1]_{6\ell}}}$ themselves, we list the
fractional difference between $g_{{}_{UV,4\ell}}$ and
$g_{{}_{UV,[1,1]_{6\ell}}}$, denoted
$\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,[1,1]_{6\ell}}})$ and the fractional
difference between $g_{{}_{UV,6\ell}}$ and $g_{{}_{UV,[1,1]_{6\ell}}}$,
denoted $\Delta(g_{{}_{UV,6\ell}},g_{{}_{UV,[1,1]_{6\ell}}})$.

##  IV Assessment of Scheme Transformations

Since $b_{3}$ and $g_{{}_{UV,6\ell}}$ are scheme-dependent, one should assess
the effect of this scheme dependence in a study of a UV zero of the beta
function for this theory. A scheme transformation can be expressed as a
mapping between $g$ and $g^{\prime}$, which we write as
$g=g^{\prime}f(g^{\prime})$, where $f(g^{\prime})$ is the scheme
transformation function, satisfying $f(0)=1$. We will consider functions
$f(g^{\prime})$ that have a Taylor series expansion

$f(g^{\prime})=1+\sum_{j=1}^{j_{\rm,max}}k_{j}\,g^{\prime j}\ ,$ (55)

where the $k_{s}$ are constants, where $j_{\rm max}$ may be finite or
infinite. The Jacobian of the transformation is $J=dg/dg^{\prime}$,

$J=1+\sum_{j=1}^{j_{max}}(j+1)k_{j}g^{\prime j}\ .$ (56)

After the scheme transformation is applied, the beta function in the new
scheme has the form (2) with $g$ replaced by $g^{\prime}$ and $b_{j}$ replaced
by $b^{\prime}_{j}$. Expressions for the $b_{j}^{\prime}$ in terms of the
$b_{j}$ and $k_{s}$ were derived in Ryttov and Shrock (2012a, b). Aside from
$b_{1}^{\prime}=b_{1}$ and $b_{2}^{\prime}=b_{2}$, these relations include

$b_{3}^{\prime}=b_{3}+k_{1}b_{2}+(k_{1}^{2}-k_{2})b_{1}\ $ (57)
$b_{4}^{\prime}=b_{4}+2k_{1}b_{3}+k_{1}^{2}b_{2}+(-2k_{1}^{3}+4k_{1}k_{2}-2k_{3})b_{1}\
,$ (58)

and so forth for $b_{j}^{\prime}$ with higher $j$.

As was discussed in Ryttov and Shrock (2012a, b) and studied further in Shrock
(2013, 2014a); Choi and Shrock (2014, 2016), in order to be physically
acceptable, a scheme transformation must satisfy several necessary conditions,
which were denoted C1 to C4. The first two conditions, C1 and C2, are that the
scheme transformation must map a real positive $g$ to a real positive
$g^{\prime}$ and should not map a moderate value of $g$, for which
perturbation theory may be reliable, to a value of $g^{\prime}$ that is so
large that perturbation theory is unreliable. The third condition, C3, is that
the Jacobian should not vanish in the region of $g$ and $g^{\prime}$ of
interest or else the transformation would be singular. A fourth condition
given in Ryttov and Shrock (2012a, b) is that, since the existence of a UV
zero of the beta function is a physical property, a scheme transformation
should satisfy the property that $\beta_{g}$ has an UV zero if and only if
$\beta_{g^{\prime}}$ has an UV zero. These conditions can easily be satisfied
by scheme transformations applied in the vicinity of a zero of the beta
function at sufficiently small coupling, but they are not automatically
satisfied, and are a significant restriction, on a scheme transformation
applied in the vicinity of a generic zero of the beta function away from the
origin. These results on scheme transformations have been applied to the study
of an IR zero in the beta function of an asymptotically free theory, such as a
non-Abelian gauge theory with a certain content of massless Dirac fermions in
$d=4$ dimensions Ryttov and Shrock (2012b); Shrock (2013, 2014a); Ryttov
(2014a, b); Gracey and Simms (2015); Choi and Shrock (2014, 2016) and the
Gross-Neveu model Gross and Neveu (1974) in $d=2$ Choi  _et al._ (2017). They
have also been applied to assess scheme dependence in probing for a possible
UV zero in the beta function of an IR-free theory such as $O(N)$
$|\vec{\phi}|^{4}$ theory in $d=4$ Shrock (2014b, 2017) (reviews include Zinn-
Justin (1989); Kleinert and Schulte-Frohlinde (2001)).

Since $\beta_{g,6\ell}$ does not have a UV zero if $N<N_{c}$, whereas
$\beta_{g,4\ell}$ has, at least formally, a UV zero for all physical $N$, a
natural method to use to study the effects of scheme dependence is to
construct a scheme transformation that eliminates the $O(g^{4})$ term in
$\beta_{g^{\prime},6\ell}$ and thus yields a beta function consisting of just
the first two (scheme-independent) terms. Since the beta function in the
transformed scheme always has a UV zero, this scheme transformation would not
satisfy condition C4. However, by applying it, one can at least gain some
information about the degree of scheme dependence in the evidence for or
against the property that, at a given
$N\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}N_{c}$,
the six-loop beta function in a particular scheme has a UV zero.

To carry out this procedure, we will use the results of Refs. Ryttov and
Shrock (2012b); Shrock (2013, 2014a), which presented scheme transformations
that can be used to set $b_{\ell}^{\prime}=0$ for $\ell\geq 3$, thereby
reducing the beta function to its two scheme-independent terms (the ’t Hooft
scheme). The simplest way to do this is to set $k_{1}=0$ in Eq. (55) and then
solve the equation $b_{3}^{\prime}=0$ for $k_{2}$, obtaining

$k_{2}=\frac{b_{3}}{b_{1}}\ .$ (59)

Although we will only need to apply the procedure here to set
$b_{3}^{\prime}=0$, since this is the highest-order coefficient that has been
calculated for this theory, we briefly review how the procedure works if one
has a beta function calculated to higher order. One next substitutes the value
of $k_{2}$ from Eq. (59) into the equation for $b_{4}^{\prime}$, Eq. (58), and
solves the equation $b_{4}^{\prime}=0$ for $k_{3}$. This procedure is applied
iteratively to solve for $k_{j}$ with $j\geq 4$ so as to render
$b_{j+1}^{\prime}=0$. At least formally, a solution is guaranteed, since the
condition that $b_{j+1}^{\prime}=0$ is a linear equation for $k_{j}$ for all
$j\geq 2$. However, while this procedure can be carried out for sufficiently
weak coupling, e.g., in the deep UV limit of a UV-free theory such as QCD, as
originally noted by ’t Hooft ’t Hooft (1979), Refs. Ryttov and Shrock (2012b);
Shrock (2013, 2014a) showed that it can be more difficult to do this with a
physically acceptable scheme transformation when studying a zero of the beta
function away from the origin in coupling constant space (an IR zero in a UV-
free theory or a UV zero in an IR-free theory).

Given that we take $k_{1}=0$ in $f(g^{\prime})$, the procedure for
constructing and applying this scheme transformation here requires only the
determination of a single parameter, $k_{2}$, since we only have to eliminate
$b_{3}^{\prime}$ to reduce the six-loop beta function to its minimal scheme-
independent first two terms in the transformed scheme. Thus, we construct a
scheme transformation with $j_{\rm max}=2$, $k_{1}=0$ and $k_{2}=b_{3}/b_{1}$,
as in Eq. (59), so as to render $b_{3}^{\prime}=0$. This is the transformation
with $f(g^{\prime})=1+(b_{3}/b_{1})g^{\prime 2}$, namely

$g=g^{\prime}\Big{[}1+\Big{(}\frac{b_{3}}{b_{1}}\Big{)}g^{\prime 2}\Big{]}\ .$
(60)

Then, since $b_{3}^{\prime}=0$, the beta function in the transformed scheme is

$\beta_{g^{\prime},6\ell}=b_{1}g^{\prime 2}+b_{2}g^{\prime 3}\ .$ (61)

Note that $b_{3}/b_{1}\sim 1/N$ for large $N$, so as $N\to\infty$, the scheme
transformation (60) approaches the identity mapping.

To check whether, for a given $N$, this scheme transformation satisfies at
least the first three conditions for acceptibility, one then calculates how
close $g$ and the corresponding $g^{\prime}$ are to each other. For a given
$N$ and resultant UV zero of the four-loop beta function, $g_{{}_{UV,4\ell}}$,
one thus solves the cubic equation (60) for $g^{\prime}$ with
$g=g_{{}_{UV,4\ell}}$, using the minimal positive real root as $g^{\prime}$.
In Table 2 we list illustrative values of $N$ and $g_{{}_{UV,4\ell}}$,
together with the solution for $g^{\prime}$ from Eq. (60) with
$g=g_{{}_{UV,4\ell}}$ and the fractional difference
$\Delta(g_{{}_{UV,4\ell}},g^{\prime})$. Since the scheme transformation (60)
approaches the identity as $N\to\infty$, it follows that
$\lim_{N\to\infty}g^{\prime}=\lim_{N\to\infty}g$ for all $g$. In particular,
as is evident from Table 2, in the region
$N\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}10^{3}$,
if one sets $g=g_{{}_{UV,4\ell}}$, then the corresponding value of
$g^{\prime}$ is close to this value. For example, for $N=2\times 10^{3}$ and
$N=10^{4}$, the respective fractional differences between $g_{{}_{UV,4\ell}}$
and the corresponding $g^{\prime}$ are approximately 8 % and 2 % in magnitude.
Furthermore, since $b_{1}$ and $b_{3}$ are both positive, the condition that
the Jacobian $J$ should not vanish is satisfied. However, we find that in the
region of
$N\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}N_{c}$,
although one can formally apply this scheme transformation, thereby switching
to a scheme in which the six-loop beta function has a UV zero, the value of
the coupling at this UV zero, $g^{\prime}$, is substantially different from
$g_{{}_{UV,4\ell}}$. Hence, in the region
$N\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}N_{c}$,
this theory does not satisfy a necessary requirement for a reliably calculable
UV zero of the beta function, namely that the values calculated in different
schemes should be close to each other. In this region of $N$, the scheme
transformation obviously also fails to satisfy the fourth condition, C4, for
acceptibility discussed above. These results are consistent with the
conclusion that if
$N\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}N_{c}$,
then the beta function, calculated to $O(g^{4}$), does not exhibit evidence
for an ultraviolet zero.

##  V Conclusions

In this paper we have investigated the ultraviolet behavior of the
$|\vec{\phi}|^{6}_{3}$ theory, focusing on the question of whether, for a
given $N$, this theory exhibits robust evidence of an ultraviolet zero in the
beta function, as calculated to the six-loop (i.e., $O(g^{4})$) order. We make
use of the result for the six-loop beta function calculated in the minimal
subtraction scheme in Hager (2002). Early work Townsend (1977); Pisarski
(1982); Appelquist and Heinz (1982) established that this theory has a UV zero
$g_{{}_{UV,4\ell}}$ in the four-loop beta function, which is reliably
calculable for large $N$. We find that the six-loop beta function from Hager
(2002) has a UV zero if $N>N_{c}$, where $N_{c}\simeq 796$. From studying the
fractional difference between $g_{{}_{UV,4\ell}}$ and $g_{{}_{UV,6\ell}}$ as a
function of $N$, we conclude that this zero in the six-loop beta function is
robust for $N$ well above $N_{c}$. To study the properties of the theory for
finite $N$ further, we have analyzed the Padé approximant to the (reduced)
six-loop beta function, $[1,1]_{6\ell}$. Although this approximant does have a
UV zero for a range of $N$ below $N_{c}$, the value of the coupling at this UV
zero, $g_{{}_{UV,[1,1]_{6\ell}}}$, is not close to the value
$g_{{}_{UV,4\ell}}$ obtained from the four-loop beta function, so that this
does not constitute evidence that the theory actually has a reliably
calculable UV zero in this range of $N$. Our application of a scheme
transformation to the minimal two-term beta function (’t Hooft scheme) yields
the same conclusion. Quantitatively, if one imposes the criterion that the
fractional difference between $g_{{}_{UV,4\ell}}$ and $g_{{}_{UV,6\ell}}$
should be smaller than, say, 15 % for the calculation of the UV zero in the
beta function to be reasonably reliable, then our results show that this
criterion is satisfied for
$N\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}}2\times
10^{3}$. Clearly, there is some arbitrariness in this benchmark value of 15 %
for the relative agreement of these couplings; imposing a more (less)
stringent requirement on the relative agreement of $g_{{}_{UV,4\ell}}$ and
$g_{{}_{UV,6\ell}}$ would shift the estimated minimal value of $N$ for a
reliable calculation to a higher (lower) value than $2\times 10^{3}$. It is
again noted that nonperturbative effects may be important for this theory.
However, we believe that it is useful at least to investigate the basic
perturbative question of the range in $N$ for which the beta function,
calculated to the highest order to which it is known, yields robust evidence
for an ultraviolet zero. We have addressed this question in the present paper.

###### Acknowledgements.

This research was partly supported by the U.S. National Science Foundation
grant NSF PHY-22-15093.

##  Appendix A Conversions Between Different Normalization Conventions

In the literature, several different normalization conventions have been used
for the interaction coupling in $|\vec{\phi}|^{6}_{3}$ theories. We list some
conversion relations here and remark on the consequences of these
normalizations for the respective beta functions. All of the works included
here used a real $N$-component scalar field $\phi$ except for Appelquist and
Heinz (1981, 1982), which used a complex $N$-component scalar field,
equivalent to a $2N$-component real field. Aside from numerical prefactors,
there have been two general classes of normalization conventions. The first
class of normalizations involves division of the coupling by $N^{2}$ in the
interaction term ${\cal L}_{\rm int.}$, while the second does not. We list the
interaction terms below, in the notation used in the original papers, with
superscripts added for clarity. Ref. Townsend (1977) by Townsend (T) used the
interaction term

${\cal L}^{(\rm T)}_{\rm int.}=\frac{1}{6!N^{2}}\eta|\vec{\phi}|^{6}\ ,$ (62)

Ref. Appelquist and Heinz (1982) by Appelquist and Heinz (AH) used the
interaction term (with a complex field $\vec{\phi}$)

${\cal L}^{(\rm AH)}_{\rm
int.}=\frac{1}{6N^{2}}g(\vec{\phi}^{\dagger}\cdot\vec{\phi})^{3}\ ,$ (63)

and Ref. Bardeen  _et al._ (1984) by Bardeen, Moshe, and Bander (BMB) used

${\cal L}^{(\rm BMB)}_{\rm int.}=\frac{1}{6N^{2}}\eta|\vec{\phi}|^{6}\ .$ (64)

Among the second class of normalizations, Ref. Pisarski (1982) by Pisarski (P)
used

${\cal L}^{(\rm P)}_{\rm int.}=\frac{\pi^{2}}{3}\lambda|\vec{\phi}|^{6}\ .$
(65)

while Ref. Hager (2002) by Hager (H) used

${\cal L}^{(\rm H)}_{\rm int.}=\frac{1}{6!}w|\vec{\phi}|^{6}\ ,$ (66)

and also the rescaling

$\bar{w}\equiv\frac{w}{32\pi^{2}}\ .$ (67)

(The reader should not confuse the sextic coupling $\lambda^{(P)}$ used in
Pisarski (1982) with the quartic coupling $\lambda$ that we have used in Eq.
(1).) We have employed the BMB normalization convention in our Eq. (1) but
with the symbol $g$ rather than $\eta$. These couplings are related to each
other as follows, where we use the notation in the original papers:

$\displaystyle\eta^{(\rm BMB)}=\frac{1}{2}g^{(\rm AH)}=\frac{1}{5!}\eta^{(\rm
T)}=2\pi^{2}N^{2}\lambda^{(\rm P)}=\frac{4\pi^{2}N^{2}}{15}\bar{w}^{(\rm H)}\
.$ (68) (69) (70)

These different normalizations affect the definition of the respective beta
functions. In general, consider two $|\vec{\phi}|^{6}$ interaction couplings
$c$ and $c^{\prime}$ that are related to each other according to

$c^{\prime}=rc\ ,$ (71)

where $r$ is a multiplicative factor, The corresponding beta functions are
$\beta_{c}=dc/d\ln\mu$ and $\beta_{c^{\prime}}=dc^{\prime}/d\ln\mu$, with
respective series expansions

$\beta_{c}=c\sum_{j=1}^{\infty}b_{c,j}c^{j}$ (72)

and

$\beta_{c^{\prime}}=c^{\prime}\sum_{j=1}^{\infty}b_{c^{\prime},j}c^{\prime j}\
.$ (73)

Then, since $b_{c,j}c^{j}=b_{c^{\prime},j}c^{\prime
j}=b_{c^{\prime},j}(rc)^{j}$, it follows that these expansion coefficients are
related according to

$b_{c^{\prime},j}=r^{-j}b_{c,j}\ .$ (74)

Consequently, as is evident in Eqs. (3), (6), and (17), with the T, AH, or BMB
normalizations of the coupling, the corresponding beta function vanishes in
the limit $N\to\infty$. Hence, if $m^{2}$ and $\lambda$ are tuned to zero, in
this limit the theory is scale-invariant, and it is this scale invariance that
was found in Bardeen  _et al._ (1984) to be spontaneously broken if the BMB
coupling is larger than $(4\pi)^{2}$. In contrast, with the normalization used
in Pisarski (1982) and Hager (2002), the respective beta functions
$\beta_{\lambda}=d\lambda/d\ln\mu$ and $\beta_{\bar{w}}=d\bar{w}/d\ln\mu$ do
not vanish for large $N$. Note that a ratio such as $b_{1}b_{3}/b_{2}^{2}$ is
invariant under these changes in normalizations.

For reference, the conversion relations for the UV zero of the four-loop beta
function are

$\displaystyle\eta^{(\rm BMB)}_{{}_{UV,4\ell}}=192\ \Leftrightarrow\ g^{(\rm
AH)}_{{}_{UV,4\ell}}=384\ \Leftrightarrow$ (75) (76)
$\displaystyle\lambda^{(\rm P)}_{{}_{UV,4\ell}}=\frac{96}{\pi^{2}N^{2}}\
\Leftrightarrow\ \bar{w}_{{}_{UV,4\ell}}=\frac{720}{\pi^{2}N^{2}}\ .$ (77)

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Table 1: From left to right, the columns of this table list (i) $N$; (ii) the
UV zero, $g_{{}_{UV,4\ell}}$, of the four-loop beta function,
$\beta_{g,4\ell}$; (iii) the UV zero, $g_{{}_{UV,6\ell}}$, of the six-loop
beta function, $\beta_{g,6\ell}$; (iv) the UV zero,
$g_{{}_{UV,[1,1]_{6\ell}}}$, of the [1,1] Padé approximant to the reduced six-
loop beta function, $\beta_{g,{\rm red.},6\ell}$; and the fractional
differences, denoted for short as (v)
$\Delta_{4\ell,6\ell}\equiv\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,6\ell}})$; (vi)
$\Delta_{4\ell,[1,1]_{6\ell}}\equiv\Delta(g_{{}_{UV,4\ell}},g_{{}_{UV,[1,1]_{6\ell}}})$;
and (vii)
$\Delta_{6\ell,[1,1]_{6\ell}}\equiv\Delta(g_{{}_{UV,6\ell}},g_{{}_{UV,[1,1]_{6\ell}}})$.
The last row lists the limiting values as $N\to\infty$. We use the standard
notation $-0.331$e-2 for $-(0.331\times 10^{-2})$, etc. The symbol “n” means
that the entry is unphysical or not relevant.

$N$ $g_{{}_{UV,4\ell}}$ $g_{{}_{UV,6\ell}}$ $g_{{}_{UV,[1,1]_{6\ell}}}$
$\Delta_{4\ell,6\ell}$ $\Delta_{4\ell,[1,1]_{6\ell}}$
$\Delta_{6\ell,[1,1]_{6\ell}}$ 1 0.2356 n n n n n 10 12.21 n n n n n 100
108.09 n n n n n 300 154.71 n 378.17 n 1.44 n 900 178.05 268.07 229.16 0.506
0.287 $-0.145$ 1.0e3 179.37 249.49 224.79 0.391 0.253 $-0.0990$ 2.0e3 185.505
210.34 207.07 0.134 0.116 $-0.0156$ 4.0e3 188.71 199.90 199.24 0.0593 0.0558
$-0.331$e-2 1.0e4 190.67 194.93 194.83 0.0223 0.0218 $-0.513$e-3 $\infty$ 192
192 192 0 0 0

Table 2: From left to right, the columns of this table list (i) $N$; (ii) the UV zero, $g_{{}_{UV,4\ell}}$, of the four-loop beta function, $\beta_{g,4\ell}$; (iii) $g^{\prime}$, the value of $g$ in the transformed scheme with $b_{3}^{\prime}=0$ obtained via the solution of Eq. (60) with $g$ set equal to $g_{{}_{UV,4\ell}}$; and (iv) the fractional difference between these values, denoted for short as $\Delta_{\rm tran.}\equiv\Delta(g_{{}_{UV,4\ell}},g^{\prime})$. The last row lists the limiting values as $N\to\infty$. We use the standard notation 1.0e3 for $1.0\times 10^{3}$, etc. $N$ | $g_{{}_{UV,4\ell}}$ | $g^{\prime}$ | $\Delta_{\rm tran.}$
---|---|---|---
100 | 108.09 | 69.901 | $-0.353$
300 | 154.71 | 116.09 | $-0.250$
900 | 178.05 | 152.90 | $-0.141$
1.0e3 | 179.37 | 155.67 | $-0.132$
2.0e3 | 185.505 | 170.50 | $-0.0809$
4.0e3 | 188.71 | 180.04 | $-0.0459$
1.0e4 | 190.67 | 186.84 | $-0.0201$
$\infty$ | 192 | 192 | 0