# Bag of Baselines for Multi-objective Joint Neural Architecture Search and
Hyperparameter Optimization

Julia Guerrero-Viu<EMAIL_ADDRESS>
Sven Hauns<EMAIL_ADDRESS>
Sergio Izquierdo<EMAIL_ADDRESS>
Guilherme Miotto<EMAIL_ADDRESS>
Simon Schrodi<EMAIL_ADDRESS>
André Biedenkapp<EMAIL_ADDRESS>
University of Freiburg
Thomas Elsken<EMAIL_ADDRESS>
Bosch Center for Artificial Intelligence
Difan Deng<EMAIL_ADDRESS>
Marius Lindauer<EMAIL_ADDRESS>
Leibniz University Hannover
Frank Hutter<EMAIL_ADDRESS>
University of Freiburg and Bosch Center for Artificial Intelligence

###### Abstract

Neural architecture search (NAS) and hyperparameter optimization (HPO) make
deep learning accessible to non-experts by automatically finding the
architecture of the deep neural network to use and tuning the hyperparameters
of the used training pipeline. While both NAS and HPO have been studied
extensively in recent years, NAS methods typically assume fixed
hyperparameters and vice versa — there exists little work on joint NAS + HPO.
Furthermore, NAS has recently often been framed as a multi-objective
optimization problem, in order to take, e.g., resource requirements into
account. In this paper, we propose a set of methods that extend current
approaches to jointly optimize neural architectures and hyperparameters with
respect to multiple objectives. We hope that these methods will serve as
simple baselines for future research on multi-objective joint NAS + HPO. To
facilitate this, all our code is available at https://github.com/automl/multi-
obj-baselines.

##  1 Introduction

Neural architecture search (NAS) (zoph-iclr17a; elsken_survey;
wistuba2019survey) and hyperparameter optimization (HPO) (feurer-
automlbook19a) make deep learning accessible to non-experts by automatically
tuning the employed neural network architecture and the hyperparameters of a
deep learning algorithm.

While both fields have been studied extensively in recent years, NAS methods
typically assume fixed hyperparameter configurations and vice versa.111HPO
methods sometimes also consider some architectural hyperparameters but often
specialize in optimizing (few) continuous hyperparameters rather than the
discrete choices characteristic of NAS. There exists little work on _jointly_
optimizing hyperparameter configurations and neural network architectures
(Zela18; dong2020autohas; ZimLin2021a), even though it seems natural that
different architectures require different hyperparameter configurations to
yield optimal performance. Indeed, there is evidence that this is actually the
case.

For example, DBLP:journals/corr/Gastaldi17 showed that the strongest version
of the proposed shake-shake regularization performs best for some
architectures, but is too strong for other architectures, resulting in poor
performance or even divergence during training. Thus, a joint optimization of
hyperparameter configurations and architecture can be expected to be
beneficial.

While NAS and HPO methods typically optimize for accuracy, in many real-world
applications there is more than one objective. Common objectives next to
accuracy are, e.g., memory requirements, energy consumption or latency on the
target hardware where the neural network is eventually deployed.

In this paper, we take a first step in the direction of _multi-objective joint
NAS + HPO_ by proposing and empirically evaluating a set of simple, yet
powerful baseline methods. All our baseline methods essentially extend current
NAS or HPO approaches to cover both classical and architectural
hyperparameters, optimized under multiple objectives; see Table 1 for an
overview of the methods we propose.

Proposed Method | Based on | Extended by
---|---|---
SH-EMOA (Sec. 3.1) | multi. obj. evolution | successive halving
MO-BOHB (Sec. 3.2) | BOHB | multi-objective for candidate selection, MOTPE
MS-EHVI (Sec. 3.3) | BO with EHVI | no surrogate for cheap objectives
MO-BANANAS (Sec. 3.4) | BANANAS | multi-objective candidate selection, successive halving
BULK & CUT (Sec. 3.5) | EA, BO | network morphism, pruning with knowledge distillation, constrained BO

Table 1: Overview of the proposed methods.

##  2 Related Work and Background

#### Neural Architecture Search (NAS).

NAS refers to the task of automatically learning neural network architectures
from data (elsken_survey). NAS approaches often employ black-box optimization
methods, such as evolutionary algorithms (Real17; real_regularized_2018),
reinforcement learning (zoph-iclr17a), or Bayesian optimization (mendoza-
automl16a; kandasamy_neural_2018; white2019bananas). However, due to the large
computational costs, researchers have developed methods tailored towards NAS,
e.g., (gradient-based) optimization on one-shot models (bender_icml:2018;
Pham18; darts).

#### Hyperparameter Optimization (HPO).

It is widely acknowledged that tuned hyperparameter configurations can improve
the performance of machine learning models. Tuning manually, however, is a
tedious and error-prone task. The field of HPO (see e.g., feurer-
automlbook19a, for an overview) automates the search for well performing
hyperparameter configurations for the data at hand. Commonly, HPO is treated
as a black-box optimization. Bayesian Optimization (BO) (brochu-arXiv10a;
shahriari-ieee16a) is a popular framework for global optimization of expensive
black-box functions and has shown great success in HPO (see e.g., snoek-
nips12a; snoek-icml14a; feurer-aaai15a; springenberg-nips2016; eriksson-
neurips19; kandasamy-jmlr20). BO models the expensive function using a cheap-
to-evaluate probabilistic surrogate model and uses an acquisition function to
trade-off exploration and exploitation for selecting a new candidate point.
Gaussian Processes (GP) (rasmussen-book06a) and expected improvement (EI)
(movckus1975bayesian) are the most common choices for these. The tree-
structured parzen estimator (TPE) (bergstra-nips11a) is an alternative
surrogate that models density functions of good and bad hyperparameter
configurations, respectively; optimizing the ratio of these densities is
equivalent to optimizing EI (bergstra-nips11a).

#### Multi-fidelity Optimization.

As fully training and evaluating ML pipelines (e.g., deep neural networks) can
be too expensive to evaluate many configurations, multi-fidelity methods
employ cheaper fidelities to reduce this cost, e.g., training only on a small
subset of the data (klein-aistats17) or for a few epochs (falkner-icml18a).
Successive Halving (SH) (jamieson2016non) and Hyperband (HB) (li-jmlr18a) are
two powerful multi-fidelity strategies that allocate more budgets on the well-
performing hyperparameter configurations and achieve strong anytime
performance. However, both strategies select new hyperparamter configuration
at random without exploiting the knowledge gained about well-performing
regions. BOHB (falkner-icml18a), which combines BO and HB, overcomes this
issue by introducing a TPE model into Hyperband to guide the search.

#### Joint NAS + HPO.

Few researchers so far have considered the joint optimization of architectures
and hyperparameter configurations. domhan-ijcai15 and mendoza-automl16a used
the random-forest-based blackbox BO method SMAC (hutter-lion11a) to jointly
optimize both architectures and hyperparameter configurations (e.g., number of
filters, number of layers/blocks, and conditional layer hyperparameters that
are only active if a layer exists). Zela18, runge2019learning and ZimLin2021a
employed the more efficient multi-fidelity method BOHB (falkner-icml18a) to
achieve the same goal. SMB19 first employ DARTS (darts) to search for better
architectures for the task of disparity estimation and then optimize the
hyperparameters of the resulting architecture with BOHB. Finally,
dong2020autohas extended NAS methods using one-shot models to also consider
hyperparameters.

#### Multi-objective Optimization

Multi-objective optimization (e.g., miettinen_evolutionary_1999) deals with
the problem of minimizing multiple objective functions
$f_{1}(\lambda),\dots,f_{n}(\lambda)$. In general, there is no single
$\lambda$ that minimizes all objectives at the same time since the objectives
are often in competition with each other. Rather, there are multiple _Pareto-
optimal_ solutions that are optimal in the sense that one cannot reduce any
$f_{i}$ without increasing at least one other $f_{j}$ ($i\neq j$). The set of
Pareto-optimal solutions is called the _Pareto front_.

#### Multi-objective Optimization with Evolutionary Algorithms.

One class of algorithms for solving multi-objective problems are evolutionary
algorithms, e.g., Deb2015. Criteria for selecting candidates being mutated and
defining the best current solutions are typically based on _non-dominated
sorting_ (NDS) (6791727; deb_fast_2002) and the _hypervolume indicator_
(10.1007/978-3-540-31880-4_5; beume2007sms; bader2011hype); we describe these
techniques in the following.

_NDS_ extends the ranking of a set of candidates based on a single objective
to multiple objectives $f=(f_{1},\dots,f_{n})$ in the following way:

  * •

Compute the Pareto front $\mathcal{F}_{1}=pareto\\_front(\mathcal{P}|f)$ of
the current population $\mathcal{P}$ and assign all members of this Pareto
front $\mathcal{F}_{1}$ the best rank.

  * •

Remove the previous Pareto front from the population $\mathcal{P}$ and compute
the Pareto front for the remaining population:
$\mathcal{F}_{2}=pareto\\_front(\mathcal{P}\char 92\relax\mathcal{F}_{1}|f)$.
Members of this new Pareto front $\mathcal{F}_{2}$ are assigned the second
best rank.

  * •

Iterate this process until all members of the population have been assigned a
rank.

Thus, a run of NDS partitions the population into sets
$\mathcal{F}_{1},\dots,\mathcal{F}_{k}$, where a candidate
$\lambda\in\mathcal{F}_{i}$ outperforms another candidate
$\lambda^{\prime}\in\mathcal{F}_{j}$ with respect to all objectives if $i<j$.

The _hypervolume indicator_ $I_{H}$ of a population measures, informally
speaking, the space of objective function values covered by the population;
maximizing the hypervolume indicator corresponds to improving the Pareto front
and finding better solutions. Based on the hypervolume indicator, the
_hypervolume subset selection problem_ (HSSP) (bader2011hype) is defined as
the problem of finding a subset $\mathcal{P}_{HSPP}\subset\mathcal{P}$ of a
certain size $k$ so that the hypervolume is maximized for this subset:
$\mathcal{P}_{HSPP}\in\operatorname*{arg\,max}_{\mathcal{P}^{\prime}\subset\mathcal{P},|\mathcal{P}^{\prime}|=k}I_{H}(\mathcal{P}^{\prime})$.
The HSSP can also be solved to identify a poorly performing candidate by
setting $k=|\mathcal{P}|-1$ and choosing the poor candidate $\lambda_{poor}$
as the one that gets removed from the population via HSSP:
$\\{\lambda_{poor}\\}=\mathcal{P}\setminus\mathcal{P}_{HSPP}$. We refer to
bader2011hype for a more formal introduction.

#### Multi-objective Optimization with Bayesian Optimization (BO).

Traditional BO approaches only consider single objectives. However, extending
BO to the multi-objective case only requires a few modifications. For example,
similar to how EI considers improvements of the objective based on a
surrogate, Expected Hypervolume Improvement (EHVI) (emmerich2005single)
considers contributions to the Pareto front based on a surrogate function.
EHVI has become a widely used acquisition function for multi-criteria problems
due to recent works reducing the computation overhead for its calculation
(yang2019multi; daulton2020differentiable). Recently, ozaki2020multiobjective
also extended TPE to multi-objective TPE (MOTPE) by using EHVI

and a multi-objective splitting criterium to estimate the densities used in
TPE.

#### Multi-objective and constrained NAS.

_Hardware-aware_ NAS has recently emerged as an important criterion for neural
architectures, since many real-world applications require efficient
architectures w.r.t., e.g., memory requirements, energy consumption or latency
on the target hardware where the neural network is eventually deployed.
Consequently, a line of research frames NAS either as a constrained
(mnasnet_tan; cai2018proxylessnas) or multi-objective (Elsken19;
lu2019nsganet; schorn2019automated) optimization problem to take this into
account, we refer to benmeziane2021comprehensive for a recent survey.

##  3 Proposed Methods

In the following, we propose five simple, yet powerful extensions of existing
HPO and NAS optimization techniques to multi-objective joint HPO + NAS.

###  3.1 SH-EMOA: Speeding up Evolutionary Multi-Objective Algorithms

The flexibility and conceptual simplicity of evolutionary algorithms make them
directly applicable to multi-objective optimization problems. For example, the
well-known SMS-EMOA from beume2007sms is an evolutionary algorithm that
evaluates the performance of each candidate based on its contribution to the
dominated hypervolume. Although effective, evolutionary algorithms tend to be
very sample-inefficient, making them too computationally expensive for many
practical applications. In order to deal with this problem, we propose SH-EMOA
to speed up evolutionary multi-objective algorithms (EMOA) by using a multi-
fidelity approach based on successive halving, see Algorithm 1.

After initializing the population, we iterate EMOA with doubling the training
budgets in each iteration, while the number of candidates is halved. Thus,
many candidates are evaluated with a small budget to cover a wide range of
solutions, while only well-performing candidates proceed to the next stage,
and are evaluated with the next higher budget and used to generate new
candidates. The population size remains constant, so to remove a candidate
from the population we first use non-dominated sorting (NDS) to identify the
subset of the population with lowest rank and then solve the hypervolume
subset selection problem (HSSP) on this lowest-rank subset to identify a
poorly performing candidate, which is removed from the population.

Input : number of function evaluations $n_{\text{fe}}^{total}$ , population
size $s_{\text{pop}}$, maximum budget $b_{max}$, number of SH iterations $n$,
objectives $f$

Output : Pareto front w.r.t. $f$

1 Generate initial population $\mathcal{P}$ of size $s_{\text{pop}}$

$b\leftarrow\lfloor b_{max}/2^{n-1}\rfloor$

/* initial budget */

$n_{\text{fe}}\leftarrow\lfloor
n_{\text{fe}}^{total}/\sum_{i=0}^{n-1}2^{-i}\rfloor$

/* number of FE for the initial budget */

2 for  _$i=1$ to $n$_ do

3 Evaluate $f(\lambda)$ for all $\lambda\in\mathcal{P}$ with budget $b$

_ _ for  _$j=1$ to $n_{\text{fe}}-s_{\text{pop}}$ _

__ /* generate candidates for remaining FEs */

4 do

Generate new candidate $\lambda_{\text{new}}$

/* parent selection and variation */

5 Evaluate $f(\lambda_{\text{new}})$ on budget $b$

6
$[\mathcal{F}_{1},...,\mathcal{F}_{k}]\leftarrow\text{NDS}(\mathcal{P}\cup\\{\big{(}\lambda_{\text{new}},f(\lambda_{\text{new}})\big{)}\\})$

7
$\lambda_{\text{poor}}\leftarrow\text{HSSP}(\mathcal{F}_{k},|\mathcal{F}_{k}|-1)$

8
$\mathcal{P}\leftarrow(\mathcal{P}\cup\\{\big{(}\lambda_{\text{new}},f(\lambda_{\text{new}})\big{)}\\})\setminus\\{\big{(}\lambda_{\text{poor}},f(\lambda_{\text{poor}})\big{)}\\}$

9

$n_{\text{fe}}\leftarrow n_{\text{fe}}/2$

/* half number of FE in next budget */

$b\leftarrow 2b$

/* double budget */

10

11return $pareto\\_front(\mathcal{P}|f)$

Algorithm 1 SH-EMOA

Even though SH-EMOA can be expected to require a high number of FEs, it is
conceptually simple, flexible, and highly parallelizable.

###  3.2 MO-BOHB: Generalization of BOHB to an Arbitrary Number of Objectives

In order to extend BOHB (falkner-icml18a) to multi-objective optimization, we
apply two changes, one related to the BO part and one related to the HB part.
Firstly, we replace the TPE (bergstra-nips11a) originally used in BOHB by
MOTPE (ozaki2020multiobjective) to guide the search by selecting new
configurations under consideration of multiple objectives. Secondly, we extend
HB (li-jmlr18a) in a similar fashion as for SH-EMOA and MOTPE to decide with
which configuration to proceed in the next stage: we use NDS and the result of
the HSSP (bader2011hype).

Our proposed MO-BOHB generalizes BOHB to an arbitrary number of objectives by
making the aforementioned modifications; in particular, a single objective is
only a special case. Note that this approach can also be used to extend to
other methods combining a (multi-objective) black-box optimizer and multi-
fidelity optimization strategy.

###  3.3 MS-EHVI: Mixed Surrogate Expected Hypervolume Improvement

Although EHVI (emmerich2005single) can be directly applied for joint NAS and
HPO obtaining competent results, we further enhance the algorithm by a simple
observation from Elsken19: while some of the objective functions are expensive
to evaluate (e.g., evaluating the accuracy is expensive since it requires
training the network first), other are cheap to evaluate (e.g., the number of
parameters). Thus, rather than relying on a surrogate model for _every_
objective function as in vanilla EHVI, we solely use surrogate models for the
expensive objective function and directly evaluate the cheap objectives. This
way, we avoid fitting surrogate models for objectives which are cheap to
evaluate anyway and avoid poor predictions of the surrogate models.

We refer to Algorithm 2 for pseudo-code. The approach is very similar to
performing BO with EHVI, but, instead of fitting surrogate models for each
considered dimension, we provide true evaluations for the cheap objectives.

Input : expensive and cheap objectives $f=(f_{\text{exp}},f_{\text{cheap}})$,
surrogate model $\hat{f}_{\text{exp}}$, number of function evaluations
$n_{\text{fe}}$

Output : Pareto front w.r.t. $f$

1 Initialize population $\mathcal{P}$ with initial observations

2 for  _$k=1$ to $n_{\text{fe}}$_ do

3 Fit surrogate model $\hat{f}_{\text{exp}}$ on $\mathcal{P}$

4 Select next candidate: $\lambda_{\text{new}}\in
argmax_{\lambda}\text{EHVI}\big{(}\lambda|\mathcal{P},\hat{f}_{\text{exp}},f_{\text{cheap}}\big{)}$

5 Evaluate $f_{\text{exp}}(\lambda_{\text{new}})$

6 Update data:
$\mathcal{P}\leftarrow\mathcal{P}\cup\Big{\\{}\big{(}\lambda_{\text{new}},f(\lambda_{\text{new}})\big{)}\Big{\\}}$

7

return $pareto\\_front(\mathcal{P}|f)$

Algorithm 2 MS-EHVI

Empirical results show how our method is able to obtain an appropriate
exploration of the Pareto front, effectively showing a superior performance
compared to vanilla EHVI, compare Figure 4 in the appendix.

###  3.4 MO-BANANAS

BANANAS (white2019bananas) uses an ensemble of neural networks for predicting
the performance of a neural network within BO in combination with a novel
path-based encoding. Given a set of already evaluated candidates, the best one
is chosen and mutated to generate new candidates. These candidates are
evaluated by means of the predictor, and the resulting predictions are used to
determine the next candidate for evaluation in combination with an acquisition
function as usual in BO.

To extend BANANAS for multi-objective optimization, we create a Pareto front
from the predictions for the objective function values and sort according to
crowd sorting (RN72005). Subsequently, the next architectures are drawn from
this ranking by means of independent Thompson sampling for evaluation, which
works best for candidate selection according to white2019bananas. In the same
way, parent architectures are selected for generating new candidates through
mutations by first forming a Pareto front and then sorting it using crowd
sorting, see Algorithm 3.

Note that we do not use the path-based encoding from white2019bananas since it
is not meaningful for our search space. Rather, we employ a simple real-valued
vector representation, which furthermore also directly allows us to include
non-architectural hyperparameters. We employ Gaussian noise for mutating
parents: We assume integer-valued hyperparameters (e.g., number of layers),
and normalize each value by dividing by the maximum value to map each
hyperparameter to the range $[0,1]$: we then add Gaussian noise. Before each
function evaluation, this continuous representation is discretized by choosing
the integer-valued hyperparameter which is closed to the mutation value after
normalization.

Input : neural predictor $\hat{f}$, number of candidates to mutate
$n_{\text{mut}}$, mutation variance $\sigma^{2}$, number of new candidates
$n_{\text{new}}$, objectives $f$

Output : Pareto front w.r.t. $f$

1 Generate initial population $\mathcal{P}$

2 for  _$i=1$ to $n$_ do

3 train neural predictor $\hat{f}$ on $\mathcal{P}$

4 sort $\mathcal{P}$ using $\text{NDS}(\mathcal{P})$ and
crowdingDistance$(\mathcal{P})$

5 choose top-$n_{\text{mut}}$ candidates from $\mathcal{P}$ and mutate by
adding noise $\eta\sim\mathcal{N}(0,\sigma^{2})$ (drawn independently for each
dimension of the candidates)

6 evaluate chosen $n_{\text{mut}}$ candidates using $\hat{f}$

7 choose top-$n_{\text{new}}$ candidates
$\lambda_{1},\dots,\lambda_{n_{\text{new}}}$ via independent Thompson sampling

8 evaluate $f(\lambda_{1}),\dots,f(\lambda_{n_{\text{new}}})$
$\mathcal{P}\leftarrow\mathcal{P}\cup\\{{\big{(}\lambda_{1},f(\lambda_{1})\big{)},\cdots,\big{(}\lambda_{n_{new}},f(\lambda_{n_{new}})\big{)}}\\}$

9

return $pareto\\_front(\mathcal{P}|f)$

Algorithm 3 MO-BANANAS

This approach can be further extended by successive halving to quickly discard
poorly performing architectures chosen by the neural predictor. In the
experimental section we employ this version of MO-BANANAS since we empirically
found it to yield improved performance.

###  3.5 BULK & CUT

BULK & CUT combines a very simple evolutionary strategy with BO. The name BULK
& CUT comes from the fact that the algorithm first looks for high accuracy
models by successively enlarging them with network morphisms
(chen2015net2net), then shrinking them using pruning techniques in combination
with knowledge distillation (hinton2015distilling). A BULK & CUT run comprises
three sequential phases:

  1. 1.

Initialization: sample random architectures and train them;

  2. 2.

Bulk-up: generate offsprings by applying network morphisms;

  3. 3.

Cut-down: prune bulked-up models.

After the initialization phase is completed, parents are selected for the
bulk-up phase. For this, we propose the Paretsilon Greedy criterion, which
combines non-dominated sorting and an $\epsilon$-greedy exploration strategy,
as described in Algorithm 4. This criterion attributes a non-zero chance for
being a parent to all models from the initialization phase. However, the
chance of selection is higher for individuals in fronts closer to the Pareto
Front. Individuals from the same front are selected with equal probability.

Input : Population $\mathcal{P}$, exploration probability $\epsilon$

Output : $\lambda\in\mathcal{P}$

1

2while  _True_ do

3 $\mathcal{F}\leftarrow pareto\\_front(\mathcal{P})$

4 if  _$rand()\leq 1-\epsilon$ or $\mathcal{F}==\mathcal{P}$ _ then

5 Sample $\lambda$ from $\mathcal{F}$

6 return $\lambda$

7

8 else

9 $\mathcal{P}\leftarrow\mathcal{P}\setminus\mathcal{F}$;

10

Algorithm 4 Paretsilon greedy

Once a parent is chosen, an offspring is generated by applying a network
morphism. Network morphisms are commonly employed as mutations in the NAS
literature since they avoid retraining from scratch by inheriting the
knowledge of the parent (Elsken17; Elsken19; schorn2019automated;
cai2017reinforcement). In our experiments, two morphism operators are
implemented: insert a convolutional layer and insert a fully-connected layer,
at random positions.

In the cut-down phase, we employ pruning techniques to shrink models from the
first two phases. We use structured pruning (Anwar_structured) in our
experiments, i.e., eliminating units from fully-connected layers and filters
from convolutional layers, instead of dropping individual weights.

The units/filters were ranked by the sum of their output weights, and those on
the bottom of the list were pruned.

We employ knowledge distillation (hinton2015distilling) for training the
shrunken models so that they match their parent’s output (Elsken19;
Prakosa2020ImprovingTA; chen_kd). Note that in the evaluation phase of these
models, there is no more training and the shrunk models are not fine-tuned on
training labels.

During all three phases, the non-architectural hyperparameters (e.g., learning
rate and weight decay) are optimized via constrained BO. The term constrained
here refers to the fact that the optimization of the acquisition function is
performed with constrains on all architectural hyperparameters, since they
have already been specified, as previously explained. Note that the surrogate
model still covers the architectural hyperparameters; thus it is still aware
of them and the surrogate model is fitted and shared across architectures.

Algorithm 5 summarizes how BULK & CUT works.

Input : time budgets $T_{1}<T_{2}<T_{3}$, exploration probability $\epsilon$,
objectives $f$

Output : set of Pareto optimal solutions

1 $\mathcal{P}\leftarrow\emptyset$ //population set

2 $t\leftarrow elapsed\\_time()$

3 while  _$t <T_{3}$_ do

4 if  _ $t\in[0,T_{1})$_ then

5 $\lambda_{\alpha}\leftarrow random\\_architecture()$

6

7 if  _ $t\in[T_{1},T_{2})$_ then

$\lambda_{\alpha}\leftarrow paretsilon\\_greedy(\mathcal{P},\epsilon)$

/* parent selection (see Alg. 4) */

8 $\lambda_{\alpha}\leftarrow network\\_morphism(\lambda_{\alpha})$

9

10 if  _ $t\in[T_{2},T_{3}]$_ then

$\lambda_{\alpha}\leftarrow paretsilon\\_greedy(\mathcal{P},\epsilon)$

/* parent selection (see Alg. 4) */

11 $\lambda_{\alpha}\leftarrow
prune\\_and\\_distill\\_knowledge(\lambda_{\alpha})$

12

$\lambda_{\beta}\leftarrow constrained\\_BO(\lambda_{\alpha})$

/* other hyperparameters */

13 evaluate $f(\lambda_{\alpha},\lambda_{\beta})$

14 Update $constrained\\_BO$ with
$\big{(}\lambda_{\alpha},\lambda_{\beta},f(\lambda_{\alpha},\lambda_{\beta})\big{)}$

15
$\mathcal{P}\leftarrow\mathcal{P}\cup\\{\big{(}\lambda_{\alpha},\lambda_{\beta},f(\lambda_{\alpha},\lambda_{\beta})\big{)}\\}$

16 $t\leftarrow elapsed\\_time()$

17

return $pareto\\_front(\mathcal{P})$

Algorithm 5 BULK & CUT

##  4 Experiments

We start by describing our experimental setup in Section 4.1 and present
results in Section 4.2.

###  4.1 Experimental Setup

Each of the proposed methods, along with random search as the simplest
baseline, is run 10 times on each dataset, with a runtime limit of 24 hours
(per run and per method) on a single RTX 2080 Ti GPU.

A maximum budget of 25 epochs for training a single configuration is defined,
although it is up to the methods to decide if they want to train with a
smaller budget to speed up the search. We target network size (by means of
number of parameters) and classification accuracy as the objectives of our
multi-objective optimization.

#### Datasets.

We used the Oxford-Flowers dataset (Nilsback06), a small dataset composed of
17 different classes with 80 examples each, to show the performance of the
proposed approaches in environments where many, cheap function evaluations are
available. All images are scaled down to a resolution of $16x16$ for
computational reasons. We also tested our methods on Fashion MNIST
(xiao2017/online). We split the datasets as follows: for Flowers, we randomly
split the data into 60% for training, 20% for validation and 20% for testing.
For Fashion-MNIST, we split the train set as defined in PyTorch into training
(80%) and validation (20%) and use the original test split for testing. Neural
network weights are always trained on training data, their performance on
validation data is used to guide hyperparameter and architecture optimization,
and the test set is only used for evaluation.

#### Search Space.

As both target datasets are composed of images as input, all architectures
start with a variable number of convolutional layers, with a variable number
of filters for each layer and a variable kernel size. All layers employ ReLU
activation functions, followed by max-pooling to reduce spatial dimensions.
Batch normalization may be chosen after each layer. After the last
convolution, a global average pooling may be applied. The feature volume is
flattened and fed to a variable-length sequence of fully connected layers,
with a variable number of neurons and using ReLU as the activation function.
After the last hidden layer, a final fully connected layers maps to the class
predictions. We always use Adam (Kingma-iclr15) to optimize neural network
weights, with a searchable learning rate and batch size. The number of filters
of the convolutional layers, the number of neurons on the hidden layers, the
learning rate, and the batch size are considered on a logarithmic scale.

We refer to Table 2 for a summary of our search space.

Hyperparameter | Range | Log scale
---|---|---
Num. convolutional layers | $\\{1,2,3\\}$ | No
Num. filters conv. layer $i$ | $[2^{4},2^{10}]$ | Yes
Kernel size | $\\{3,5,7\\}$ | No
Batch normalization | $\\{true,false\\}$ | No
Global average pooling | $\\{true,false\\}$ | No
Num. fully connected layers | $\\{1,2,3\\}$ | No
Num. neurons FC layer $i$ | $[2^{1},2^{9}]$ | Yes
Learning rate | $[10^{-5},10^{0}]$ | Yes
Batch size | $[2^{0},2^{9}]$ | Yes
Table 2: Joint space of architectural and non-architectural hyperparameters
being optimized.

(a) Sampled configurations for each method on Flowers. We only show a random
subset of $10\%$ of all sampled points for visualization purposes.

(b) Sampled configurations on Fashion-MNIST dataset.

Figure 1: Sampled configurations for each method on Fashion-MNIST.

###  4.2 Results

#### Visualizing sampled configurations.

Figure 1 visualizes the sampled configurations for each method across all 10
random seeds.

For the Flowers dataset (upper row), all methods (except random search)
explore the Pareto front, which is what they were designed for. However, they
also significantly differ in the exploration strategy. SH-EMOA explores both
objective functions equally well but still samples poor configurations in
later iterations. MO-BOHB tends to focus on smaller networks in later
iterations, while MS-EHVI very quickly discovers networks close to the Pareto
front and mostly samples new configures there. After an initial phase, BULK &
CUT also mostly samples candidates close to the Pareto front in later
iterations. For Fashion-MNIST, already very small networks yield a high
accuracy, making it hard to actually trade-off the different objective
functions. However, one can still see that all methods focus on promising
regions in objective function space.

#### Anytime performance over the course of multi-objective joint NAS+HPO.

Figure 2 shows the hypervolume indicator over time, averaged across the 10
independent runs per method on each of the two datasets. All our methods
clearly outperform random search. For Flowers, MS-EHVI converges very fast but
is eventually outperformed by BULK & CUT, which however performs less strongly
in the initial phase. SH-EMOA, MO-BOHB and MO-BANANAS perform similarly. For
Fashion-MNIST, MS-EHVI and BULK & CUT again slightly outperform the remaining
approaches.

#### Final results.

Figure 3 shows the final Pareto fronts for all methods when combining the
results from all seeds.

The proposed methods perform similarly for the range of parameters from
$10^{3}$ to $10^{5}$, but some methods have problems with covering smaller or
larger models. When looking at results for each seed other runs (please refer
to Figures 5 and 6 in the appendix), we however also noticed that the results
vary across seeds, indicating that initializing might have a high impact on
the performance and that the budget of 24 hours might not be sufficient for
the methods to converge or that methods simply get stuck in a local optimum.
Table 3 summarizes the obtained hypervolume of the final Pareto front for each
method.

(a) Flowers dataset on validation split.

(b) Fashion-MNIST dataset on validation split.

(c) Flowers dataset on test split.

(d) Fashion-MNIST dataset on test split.

Figure 2: Hypervolume obtained by the different methods over time. We show
means $\pm$ standard errors of the mean based on 10 independent runs of each
method.

(a) Pareto fronts on Flowers dataset.

(b) Pareto fronts on Fashion-MNIST dataset.

Figure 3: Pareto fronts by combining the results of the 10 different runs for
each method on the test split

.

| Hypervolume (mean $\pm$ std. error)
---|---
| Flowers | Fashion-MNIST
Random search | 299.05 $\pm$ 1.19 | 393.86 $\pm$ 6.97
SH-EMOA | 321.90 $\pm$ 3.79 | 409.61 $\pm$ 12.61
MO-BOHB | 317.98 $\pm$ 2.48 | 441.86 $\pm$ 7.58
MS-EHVI | 323.72 $\pm$ 2.46 | 479.13 $\pm$ 4.15
MO-BANANAS | 319.11 $\pm$ 3.59 | 400.71 $\pm$ 9.00
BULK & CUT | 329.54 $\pm$ 1.41 | 465.19 $\pm$ 3.83
Table 3: Final hypervolume obtained by each method on both test datasets. We
show means $\pm$ standard errors based on 10 independent runs.

##  5 Conclusions

We addressed the problem of joint hyperparameter optimization and neural
architecture search under multiple objectives by extending existing methods to
this scenario. We recommend that the proposed methods serve as baselines for
future research in this direction. To facilitate this, all our code is
available under https://github.com/automl/multi-obj-baselines under a
permissive Open Source license (Apache 2.0).

Acknowledgments

The authors acknowledge funding by the Robert Bosch GmbH. A part of this work
was supported by the German Federal Ministry of Education and Research (BMBF,
grant RenormalizedFlows 01IS19077C).

##  A Full Details on the Various Baselines

###  A.1 Implementation details on SH-EMOA

  * •

Population size and total number of samples: In Flowers dataset, we use
$s_{\text{pop}}=100$ and $n_{\text{fe}}^{total}=15000$. In Fashion-MNIST, we
use $s_{\text{pop}}=10$ and $n_{\text{fe}}^{total}=150$.

  * •

Parent selection: We use tournament selection by randomly sampling $k$
potential parents from the current population (uniform distribution) and
choose the parent with highest fitness. We use $k=3$.

  * •

Variation: On each step we choose either mutation or recombination strategy
with equal probability. Mutation is defined as a uniformly distributed random
variation of $5$ hyperparameters from the parent configuration. For
recombination, we use two parents and choose each hyperparameter from one of
them with equal probability. As we have a conditional search space,
relationships between hyperparameters are taken into account when a new
individual is created. For example, if a mutation increases the total number
of convolutional layers, the size of the kernel for each new layer is also
added.

###  A.2 Implementation details on MO-BOHB

Algorithms 6 and 7 show pseudo code for MO-BOHB and its sampling step,
respectively. Note the close resemblance to the original BOHB (differences
marked in red), as MO-BOHB generalizes BOHB to any number of objectives. Note
however, that there are two minor differences between the current version of
our proposed MO-BOHB and the original BOHB implementation: (i) MO-BOHB uses an
hierarchy of one-dimensional KDEs, whereas BOHB use a single multi-dimensional
KDE, and (ii) we do not multiply bandwidths by a constant factor $b_{w}$. In
future versions of MO-BOHB, we suspect that using a single multi-dimensional
KDE and multiplication of bandwidths may further improve performance by better
handling interaction effects between (architectural) hyperparameters and
encouraging more exploration around promising configurations, respectively.
Input : budgets $b_{min}$ and $b_{max}$, configurations discarding factor
$\eta\in\mathbb{N}_{>0}$, and objectives $f$ Output : Pareto front w.r.t. f  1
$s_{max}\leftarrow\lfloor\log_{\eta}\frac{b_{max}}{b_{min}}\rfloor$ 2
$\mathcal{P}_{b}\leftarrow[~{}]\forall b\in\\{\eta^{-s}\cdot
b_{max}|s=s_{max},s_{max-1},...,0\\}$ 3 while  _not stopping criterion_ do 4
for  _$s\in\\{s_{max},s_{max-1},...,0\\}$_ do 5 sample
$n=\lceil\frac{s_{max}+1}{s+1}\eta^{s}\rceil$ configurations
$\lambda_{1},...,\lambda_{n}$ using Algorithm 7 6 run modified SH on
$\lambda_{1},...,\lambda_{n}$ with initial budget $\eta^{-s}\cdot b_{max}$ 7
add observations $\\{(\lambda_{i},f(\lambda_{i}))\\}$ of each budget $b$ to
$\mathcal{P}_{b}$ 8 9 return $pareto\\_front(\mathcal{P}_{b_{max}}|f)$
Algorithm 6 MO-BOHB Input : observations $\mathcal{P}$, fraction of random
runs $\rho$, quantile $\gamma$, number of samples $n$, and minimum number of
points $N_{min}$ to build a model  Output : next configuration to evaluate  1
if  _$rand() <\rho$_ then 2 return random configuration  3 4
$b\leftarrow\operatorname*{arg\,max}\\{\mathcal{P}_{b}:|\mathcal{P}_{b}|\geq
N_{min}+2\\}$ 5 if  _$b=\emptyset$_ then 6 return random configuration  7 8
greedily split $\mathcal{P}$ into good $\mathcal{P}_{l}$ or bad
$\mathcal{P}_{g}$ observations using NDS & HSSP 9 fit KDEs $l$ and $g$ based
on $\mathcal{P}_{l}$ or $\mathcal{P}_{g}$, respectively  10 draw $n$ samples
according to $l(\lambda)$ 11 return sample with highest ratio
$\frac{l(\lambda)}{g(\lambda)}$ Algorithm 7 Sampling in MO-BOHB

In all our experiments, we set the meta-parameters of MO-BOHB as follows: For
the HB part of MO-BOHB, we set the configuration discarding factor of to
$\eta=3$, use minimum budget $b_{min}=5$ and maximum budget $b_{max}=25$. In
the BO part of MO-BOHB, we use an random fraction $\rho=1/6$, set the quantile
to $\gamma=0.1$, sample $n=24$ configurations, and use a minimum of
$N_{min}=2\cdot|HPs|+1$ points before building a model.

###  A.3 Details on MS-EHVI

Figure 4 shows a comparison of MS-EHVI to standard EHVI.

Figure 4: Comparison between EHVI and MS-EHVI on Fashion-MNIST. Hypervolume
obtained by both methods over time. We show mean $\pm$ standard error of the
mean based on 10 independent runs of each method.

##  B Supplemental Results

Figures 5 and 6 show all the Pareto fronts found with different seeds.

Figure 5: Pareto fronts obtained for different initial random seeds on Flowers
dataset.

Figure 6: Pareto fronts obtained for different initial random seeds on
Fashion-MNIST.


# On sinh-Gordon Thermodynamic Bethe Ansatz and fermionic basis.

S. Negro  SN Department of mathematical sciences, University of Durham,
Science Laboratories, South Rd, Durham DH1 3LE, United Kingdom. Dip. di Fisica
and INFN, Università di Torino, Via P. Giuria 1, 10125 Torino, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We review the construction of the fermionic basis for sinh-Gordon model and
investigate numerically the ultra-violet limit of the one-point functions. We
then compare the predictions obtained from this formalism against previously
established results.

##  1\. Introduction

In the study of a Quantum Field Theory (QFT), the one-point functions play a
fundamental rôle; indeed, when using the Operator Product Expansion (OPE) to
calculate the ultraviolet asymptotics of a correlation function, one needs to
know both the coefficients of the said expansion and the one-point functions
of the local operators in the theory. While the former are purely ultra-violet
objects and can, in principle, be extracted via perturbation theory of the
corresponding ultra-violet Conformal Field Theory (CFT), the one-point
functions depend essentially on the infra-red structure of the theory, where
perturbative techniques are of no help at all. Thus the development of new
methods to explore the infra-red region is of primary importance.

Integrable models are the perfect playground where one can experiment with new
analytical methods aimed at extracting data; in particular the sinh-Gordon
model is the simplest example of massive integrable QFT and, at the same time,
is complicated enough to display interesting structures. Moreover this model,
along with its twin, the sine-Gordon model, has received plenty of attention
in the last 30 years and nowadays most of its features are known.

Computing one-point functions in an integrable deformation of a CFT is
anything but an easy task and we wish to explain the reasons for this fact
clearly. Although in our deformed theory the conformal invariance is broken,
the local fields retain a one-to-one correspondence with those of the original
CFT, which are organized according to the corresponding Virasoro algebra in
the usual way. This means that in the perturbed theory there exist fields
$\Phi_{a}(z,\bar{z})=e^{a\eta(z,\bar{z})}$ which can be deemed as _primary_ ,
whose space of descendants can be identified with the tensor product of Verma
modules $\mathcal{V}_{a}\otimes\bar{\mathcal{V}}_{a}$ of the unperturbed CFT.
The operators acting in the space of states of the perturbed CFT can thus be
interpreted as operators acting on the corresponding Verma modules and,
consequently, one-point functions appear to be functionals on the tensor
product $\mathcal{V}_{a}\otimes\bar{\mathcal{V}}_{a}$. However, we still have
not taken in account the integrable structure of the model; in fact _all one-
point functions of descendants built out of integrals of motion identically
vanish_. This means that the correct space on which the one-point function
should be defined as a linear functional is the tensor product
$\mathcal{V}^{\textrm{quo}}_{a}\otimes\bar{\mathcal{V}}^{\textrm{quo}}_{a}$ of
the two quotient spaces

(1.1)
$\mathcal{V}^{\textrm{quo}}_{a}\doteq\mathcal{V}_{a}\Bigg{/}\sum_{k=1}^{\infty}\mathbf{i}_{2k-1}\mathcal{V}_{a}\
,\qquad\bar{\mathcal{V}}^{\textrm{quo}}_{a}\doteq\bar{\mathcal{V}}_{a}\Bigg{/}\sum_{k=1}^{\infty}\bar{\mathbf{i}}_{2k-1}\bar{\mathcal{V}}_{a}\;,$

where with $\mathbf{i}_{2k-1}$ (respectively $\bar{\mathbf{i}}_{2k-1}$) we
denote the action of the chiral (antichiral) integrals of motion on the Verma
module.

It’s now becoming clear what is the main issue: the basis we introduced above,
composed of the primary fields $\Phi_{a}(z,\bar{z})$ and their “conformal”
descendants, is a basis for the full Verma module! In order to reduce this
last to a basis of the quotient space, one has to factor out by hand all the
null vectors which arise from the action of the integrals of motion and their
form quickly becomes rather involved. One would rather work directly in the
quotient space, where the factoring of null vectors is automatically taken in
account, and fix uniquely a basis by means of some physical requirement. A
basis of this kind was actually discovered some years ago for the six-vertex
model [1, 2, 3] and immediately extended to CFT [4], sine-Gordon [5, 6, 7] and
sinh-Gordon models [8].

The building blocks of this basis are the primary fields $\Phi_{a}(z,\bar{z})$
and creation operators which, acting on the former, produce the descendants,
much like what happens for the usual conformal basis; the peculiar fact is
that these creation operators are _fermions_. There are two of them for each
chirality : $\boldsymbol{\beta}^{\ast}_{2j-1}$,
$\boldsymbol{\gamma}^{\ast}_{2j-1}$, $\bar{\boldsymbol{\beta}}^{\ast}_{2j-1}$
and $\bar{\boldsymbol{\gamma}}^{\ast}_{2j-1}$. In the above-cited articles,
these fermions were defined, in a mathematically rigorous fashion for six-
vertex, CFT and sine-Gordon models and as an educated conjecture for the sinh-
Gordon model, and their properties were thoroughly analysed; in particular for
sin(h)-Gordon model111Here and in the following, the shorthand sin(h)-Gordon
is used to denote both sine-Gordon and sinh-Gordon., the quotient space
$\mathcal{V}^{\textrm{quo}}_{a}\otimes\bar{\mathcal{V}}^{\textrm{quo}}_{a}$
was shown to allow the following basis:

(1.2)
$\boldsymbol{\beta}^{\ast}_{I^{+}}\bar{\boldsymbol{\beta}}^{\ast}_{\bar{I}^{+}}\bar{\boldsymbol{\gamma}}^{\ast}_{\bar{I}^{-}}\boldsymbol{\gamma}^{\ast}_{I^{-}}\Phi_{a}(0)\
,\qquad\mathfrak{C}(I^{+})=\mathfrak{C}(I^{-})\ ,\
\mathfrak{C}(\bar{I}^{+})=\mathfrak{C}(\bar{I}^{-})\;,$

where $I^{\pm}=\\{2i_{1}^{\pm}-1,\ldots,2i_{n}^{\pm}-1\\}$ and similarly for
$\bar{I}^{\pm}$. The symbol $\mathfrak{C}(I)$ stands for the _cardinality_ of
the set $I$ and the following multi-index notation is introduced:

(1.3) $A_{I}=A_{i_{1}}A_{i_{2}}\ldots A_{i_{n}}\
;\quad|I|\doteq\sum_{p=1}^{\mathfrak{C}(I)}i_{p}\ ;\quad
aI+b=\\{ai_{1}+b,\ldots,ai_{n}+b\\}\;,$

where $a,b\in\mathbb{Z}$ and $I=\\{i_{1},\ldots,i_{n}\\}$.

While the rigorous construction of this basis, presented in Refs. [4]-[6],
might appear somewhat cumbersome and hard to understand, when the dust raised
by their construction has fallen, the fermions reveal their true strength in
the simple and beautiful determinant formula for the one-point functions:

(1.4)
$\frac{\langle\boldsymbol{\beta}^{\ast}_{I^{+}}\bar{\boldsymbol{\beta}}^{\ast}_{\bar{I}^{+}}\bar{\boldsymbol{\gamma}}^{\ast}_{\bar{I}^{-}}\boldsymbol{\gamma}^{\ast}_{I^{-}}\Phi_{a}(0)\rangle_{R}}{\langle\Phi_{a}(0)\rangle_{R}}=\mathcal{D}\Big{(}I^{+}\cup(-\bar{I}^{+})\Big{|}I^{-}\cup(-\bar{I}^{-})\Big{|}\alpha\Big{)}\;,$

where, for two sets $A=\\{a_{j}\\}_{j=1}^{n}$ and $B=\\{b_{j}\\}_{j=1}^{n}$,
the function $\mathcal{D}$ is defined as follows

$\displaystyle\mathcal{D}(A|B|\alpha)$
$\displaystyle\doteq\left(\prod_{\ell=1}^{n}\frac{\textrm{sgn}(a_{\ell})\textrm{sgn}(b_{\ell})}{\pi}\right)\times$
(1.5)
$\displaystyle\times\det\Big{[}\Theta(ia_{j},ib_{k}|\alpha)-\pi\textrm{sgn}(a_{j})t_{a_{j}}(\alpha)\;\delta_{a_{j},-b_{k}}\Big{]}_{j,k=1}^{n}$

and the functions $\Theta(a,b|\alpha)$ and $t_{a}(\alpha)$ will be defined
below. The parameter $\alpha$ is related to the conformal dimension of the
primary field by

(1.6) $\alpha=\frac{2}{b+b^{-1}}a\;.$

A very important property of the fermions is that, aside from allowing the
construction of the descendants, they can be used in order to _shift_ the
primary and descendant fields in their conformal dimension $a$. As it is shown
in Ref. [6], if we give up the conditions
$\mathfrak{C}(I^{+})=\mathfrak{C}(I^{-}),\
\mathfrak{C}(\bar{I}^{+})=\mathfrak{C}(\bar{I}^{-})$ in favour of the less
restraining
$\mathfrak{C}(I^{+})-\mathfrak{C}(I^{-})=\mathfrak{C}(\bar{I}^{-})-\mathfrak{C}(\bar{I}^{+})=m$,
then the following relation holds

$\displaystyle\boldsymbol{\beta}^{\ast}_{I^{+}}\bar{\boldsymbol{\beta}}^{\ast}_{\bar{I}^{+}}\bar{\boldsymbol{\gamma}}^{\ast}_{\bar{I}^{-}}\boldsymbol{\gamma}^{\ast}_{I^{-}}\Phi_{a-mb}(0)\cong$
(1.7)
$\displaystyle\cong\frac{C_{m}(a)}{\prod_{j=1}^{m}t_{2j-1}(a)}\boldsymbol{\beta}^{\ast}_{I^{+}+2m}\bar{\boldsymbol{\beta}}^{\ast}_{\bar{I}^{+}-2m}\bar{\boldsymbol{\gamma}}^{\ast}_{\bar{I}^{-}+2m}\boldsymbol{\gamma}^{\ast}_{I^{-}-2m}\boldsymbol{\beta}^{\ast}_{I_{\textrm{odd}}(m)}\bar{\boldsymbol{\gamma}}^{\ast}_{I_{\textrm{odd}}(m)}\Phi_{a}(0)$

where $I_{\textrm{odd}}(m)=\\{1,3,\ldots,2m-1\\}$ and we use the symbol
$\cong$ to denote identification in weak sense (that is, under expectation
value).

As was mentioned above, for the sine-Gordon model the fermionic basis can be
build in a mathematically rigorous fashion; the authors of Ref. [6] performed
this task by relying on the fact that sine-Gordon model allows for a lattice
regularization in the form of the eight-vertex model, which is well studied
and relatively easy to manage. Conversely, for its twin, the sinh-Gordon
model, the situation is not so simple: the lattice regularization, in this
case, takes the form of a much more complicated model, where the Boltzmann
weights are defined in terms of the R-matrix of the tensor product of two
infinite-dimensional representations of $U_{q}(\widehat{\mathfrak{sl}}_{2})$
[9, 10]. So far the status of the phase transition for this model has not been
clarified and, thus, relying on the lattice regularization is not a viable
strategy for sinh-Gordon model.

An alternative approach is to start directly from the Thermodynamic Bethe
Ansatz (TBA) equations which, for the sinh-Gordon model, exhibit a very simple
structure, given the fact that the spectrum of the theory consists of a single
particle. This fact let the authors of Ref. [8] straightforwardly define all
the functions involved in the formula (1.4). However that same formula, along
with the very existence of the fermionic basis, had to be introduced as a
conjecture based on two facts:

  * •

From purely algebraic point of view, the ultra-violet (UV) limit of the sinh-
Gordon model corresponds to the CFT considered in Ref. [3]; this last theory
is, at the same time, the UV limit of the sine-Gordon model.

  * •

There are two possible interpretations of the sin(h)-Gordon action, as a
perturbation of the free boson CFT222Here and in the following the notation
$\partial=\frac{\partial}{\partial z}$ and
$\bar{\partial}=\frac{\partial}{\partial\bar{z}}$ will be used.:

$\mathcal{A}=\int\left\\{\left[\frac{1}{4\pi}\partial\eta(z,\bar{z})\bar{\partial}\eta(z,\bar{z})\right]+\frac{2\boldsymbol{\mu}^{2}}{\sin(\pi
b^{2})}\cosh[b\,\eta(z,\bar{z})]\right\\}\frac{dz\wedge d\bar{z}}{2}\;,$

or as a perturbation of the Liouville model, conventionally identified as the
minimal CFT with central charge $c=1+6Q^{2}$, where $Q=b+b^{-1}$:

$\mathcal{A}=\int\left\\{\left[\frac{1}{4\pi}\partial\eta(z,\bar{z})\bar{\partial}\eta(z,\bar{z})+\frac{\boldsymbol{\mu}^{2}}{\sin(\pi
b^{2})}e^{b\,\eta(z,\bar{z})}\right]+\frac{\boldsymbol{\mu}^{2}}{\sin(\pi
b^{2})}e^{-b\,\eta(z,\bar{z})}\right\\}\frac{dz\wedge d\bar{z}}{2}\;.$

This twofold interpretation of the action led the authors of Ref. [11] to some
functional relations for the one-point functions of sine-Gordon model, which
were named _reflection relations_. In Ref. [12] it was shown how the fermionic
basis can be interpreted as a basis of the space of states for which these
reflection relations are trivially satisfied.

A remark about the choice for the normalization of the dimensional constant is
necessary. As discussed in Ref. [8], this choice, aside from being extremely
convenient for the calculations, encloses serious physical reasons. Firstly it
takes automatically into account the change of sign in the potential energy
when passing from sinh- to sine-Gordon and encodes also the pole at $b=i$ of
this last333Due to the fact that the perturbing operator becomes irrelevant
for $b^{2}<-1$. Note that there are poles also for $b\in\mathbb{Z}$ which look
natural once one consider the physical scale of the model, namely the mass of
the particle [13].; more importantly, this normalisation allows the mass m of
both the sinh-Gordon particle and that of the sine-Gordon lowest breather to
be expressed by a universal formula:

(1.8)
$\boldsymbol{\mu}\Gamma(1+b^{2})=\left[\frac{m}{4\sqrt{\pi}}\Gamma\left(\frac{1}{2(1+b^{2})}\right)\Gamma\left(1+\frac{b^{2}}{2(1+b^{2})}\right)\right]^{1+b^{2}}\;.$

Since, for the sinh-Gordon model, the formula (1.4) and the existence of the
fermionic basis still retain the status of conjectures, it is of utmost
importance to obtain a posteriori confirmations of their validity, by checking
the predictions against known results. Analytic comparison with results of
Refs. [14] and [15] were already performed in Ref. [8].

The purpose of this paper is to obtain further confirmations of the validity
of (1.4), by means of numerical simulations. In particular the one-point
functions of the sinh-Gordon model, defined on a cylinder of radius $2\pi R$,
were numerically evaluated for very small values of the radius $R\sim 0$,
limit in which the model approaches its UV limit; these numerical results were
then compared against the theoretical behaviours obtained in Refs. [4] and
[15]. As in the last-cited article a rescaling of the model to a circumference
of fixed radius $2\pi$ is to be performed; this amounts to a renormalisation
of the physical mass $m\rightarrow mR$, so that $\boldsymbol{\mu}\propto
R^{1+b^{2}}$.

It has to be noted that, since the goal is to compare the results obtained
from (1.4) with the known “CFT behaviour”, it is wise to avoid the possible
complications arising in the regions of the parameter space where $a-b<0$. Let
us clarify this point.

Looking at the formula for the conformal dimension shift of the fields (1.7)
we see that the ratio of expectation values of the two primary fields
$\Phi_{a-b}(0)$ and $\Phi_{a}(0)$ can be expressed in terms of the ratio of
the one-point function of the descendant
$\boldsymbol{\beta}_{1}^{\ast}\bar{\boldsymbol{\gamma}}_{1}^{\ast}\Phi_{a}(0)$
with that of the primary field $\Phi_{a}(0)$; in formulae

(1.9)
$\frac{\langle\Phi_{a-b}(0)\rangle}{\langle\Phi_{a}(0)\rangle}=\frac{C_{1}(a)}{t_{1}(a)}\frac{\langle\boldsymbol{\beta}_{1}^{\ast}\bar{\boldsymbol{\gamma}}_{1}^{\ast}\Phi_{a}(0)\rangle}{\langle\Phi_{a}(0)\rangle}\;.$

As one approaches the UV limit $R\sim 0$, the one-point functions of primary
fields are believed to behave as three-point functions of the Liouville CFT
[15] with two additional fields, of dimensions
$\Delta_{\pm}=\frac{Q^{2}}{4}-P(R)^{2}$ where $P(R)$ is the quantized momentum
of Liouville CFT, placed at $\pm\infty$. This, however, holds true only if the
dimensions of the fields are positive, which means

(1.10) $\left\\{\begin{array}[]{l}0<a<Q\\\
0<a-b<Q\end{array}\right.\quad\Rightarrow\quad b<a<Q\;.$

This fact becomes evident, for example, sending $a\rightarrow 0$; in this
case, the expectation value of the field $\Phi_{-b}(0)=e^{-b\eta(0)}$ can be
calculated directly in terms of the ground-state energy
$E(R)\underset{R\rightarrow 0}{\sim}-\frac{\pi}{6R}c_{\textrm{eff}}(R)$ where
[15]

(1.11) $c_{\textrm{eff}}(R)\underset{R\rightarrow
0}{\sim}1-\frac{24\pi}{\Big{(}\delta_{1}-4Q\log\frac{R}{2\pi}\Big{)}^{2}}\;,$

and $\delta_{1}$ is a constant. With (1.4), (1.9) and the relation between the
function $\Theta(i,-i|0)$ and $E(R)$ shown in Ref. [6], one obtains:

(1.12) $\langle e^{-b\eta(0)}\rangle=-\frac{C_{1}(0)}{\pi
m^{2}t_{1}(0)}\left(\frac{1}{R}+\frac{d}{dR}\right)E(R)\underset{R\rightarrow
0}{\sim}\frac{\pi c_{1}(0,b)}{2m^{2}Q^{2}}\frac{R^{-2(b^{2}+1)}}{(-\log
R)^{3}}\;,$

where the definition (LABEL:eq:C-constants) was applied, setting
$C_{1}(a)/t_{1}(a)\underset{R\rightarrow 0}{\sim}c_{1}(a,b)R^{2b(2a-b)}$,
$c_{1}(a,b)$ being a function of $a$ and $b$ only. On the other hand, using
the formula for the Liouville three-point amplitude (4.63) found in Refs. [16]
and [17], the result is radically different

(1.13) $\displaystyle\langle
e^{-b\eta(0)}\rangle=\frac{\langle\Phi_{\frac{Q}{2}-P}(-\infty)|\Phi_{-b}(0)|\Phi_{\frac{Q}{2}+P}(\infty)\rangle}{\langle\Phi_{\frac{Q}{2}-P}(-\infty)|\Phi_{\frac{Q}{2}+P}(\infty)\rangle}\underset{R\rightarrow
0}{\sim}k(a,b)R^{2(1+b^{2})}\;.$

It is clear that outside the natural region $b<a<Q$, the sinh-Gordon model do
no more approaches naïvely the Liouville CFT: there are contributions not
taken in account which become important. However, as said above, rather than
exploring the UV limit of the sinh-Gordon model per se, the goal of this paper
is to use it in order to obtain evidence of the agreement between the
predictions obtained from the fermionic basis and the results known in the
literature: for this reason from now on the parameter space will be restricted
to the region $0<b<a<Q$.

##  2\. The fermionic basis

Let us review briefly the properties of the fermionic basis.
The two-fold interpretation of the sinh-Gordon action that we mentioned above
has an interesting and important consequence. If we look at sinh-Gordon as a
deformation of the free boson CFT, then the natural choice for the descendants
of the primary field $\Phi_{a}(0)=e^{a\eta(0)}$ are normal ordered products of
$e^{a\eta(0)}$ with polynomials of even degree444We limit ourselves to even
degree polynomials, since $\mathcal{V}_{a}^{\textrm{quo}}$ non-trivial
subspaces are of even dimension only. in the derivatives of $\eta(0)$. In this
_Heisenberg basis_ the one-point functions inherit the natural free boson
symmetry

(2.14) $\sigma_{1}\;:a\rightarrow-a\;.$

On the other hand, if we consider the sinh-Gordon model as a deformation of
the Liouville CFT, the descendants of $\Phi_{a}(0)$ are more naturally defined
as normal-ordered products of $e^{a\eta(0)}$ with polynomials in even-degree
derivatives of $T(z,\overline{z})$ and $\bar{T}(z,\overline{z})$, where

$\displaystyle T(z,\overline{z})\doteq
T_{z,z}(z,\overline{z})=-\frac{1}{4}\Big{[}\partial\eta(z,\overline{z})\Big{]}^{2}+\frac{Q}{2}\partial^{2}\eta(z,\overline{z})\>,$
$\displaystyle\bar{T}(z,\overline{z})\doteq
T_{\overline{z},\overline{z}}(z,\overline{z})=-\frac{1}{4}\Big{[}\bar{\partial}\eta(z,\overline{z})\Big{]}^{2}+\frac{Q}{2}\bar{\partial}^{2}\eta(z,\overline{z})\>,$

are the components of Liouville energy-momentum tensor. It is natural to
assume that in this basis, that we call the _Virasoro basis_ , the one-point
functions retain the symmetry of the Liouville model

(2.16) $\sigma_{2}\;:a\rightarrow Q-a\;.$

Since both the Heisenberg and the Virasoro basis are, when the action of the
integrals of motion has been factored out, fully fledged bases of sinh-Gordon
space of states, the one-point functions in any possible basis have to
transform in some definite way under the symmetries $\sigma_{1}$ and
$\sigma_{2}$. This fact gives rise to the above-mentioned reflection relations
and suggests that there must exist a basis in which both these symmetries act
in a simple, multiplicative way: this particular basis is the _fermionic
basis_ ; we consider then the fermions as defined by their behaviour under the
symmetries $\sigma_{1}$ and $\sigma_{2}$. Starting from the Liouville CFT,
where the fermions $\boldsymbol{\beta}^{\textrm{CFT}\,\ast}$ and
$\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}$ can be defined as an intrinsic
property of the model [8], we see that the reflections act on the fermionic
basis as follows:

$\displaystyle\phantom{\sigma_{1}\;:\;}\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow
u(a)\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\phantom{\qquad,\qquad\sigma_{2}\;:\qquad\;}\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}$
(2.17)
$\displaystyle\sigma_{1}\;:\;\phantom{\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow
u^{-1}(-a)\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}}\qquad,\qquad\sigma_{2}\;:\;$
$\displaystyle\phantom{\sigma_{1}\;:\;}\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow
u^{-1}(-a)\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}\phantom{\qquad,\qquad\sigma_{2}\;:\;\;}\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}$

where

(2.18)
$u(a)\doteq\frac{-2a+b(2m-1)}{2a+b^{-1}(2m-1)}=\frac{-Q\alpha+b(2m-1)}{Q\alpha+b^{-1}(2m-1)}$

and for the second chirality we only have to change $a$ in $-a$ in the above
function. There is an additional symmetry which was considered in Ref. [8],
that is the duality $b\rightarrow b^{-1}$, under which our fermions simply
exchange

$\displaystyle\phantom{\textrm{duality}\quad\;}\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}$
(2.19) $\displaystyle\textrm{duality}\;:$
$\displaystyle\phantom{\textrm{duality}\quad\;}\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}$

The normalization of the fermions is such that when expressing the descendants
in fermionic basis in terms of Virasoro descendants we have

(2.20)
$\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{I^{+}}\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{I^{-}}\Phi_{a}=C_{I^{+},I^{-}}\left\\{\mathbf{l}^{n}_{-2}+\cdots\right\\}\Phi_{a}\;,\qquad\mathfrak{C}(I^{+})=\mathfrak{C}(I^{-})=n\;,$

with $\mathbf{l}_{n}$ being the coefficients of the Laurent expansion of the
Liouville energy-momentum tensor component $T(z,\overline{z})$ while
$C_{I^{+},I^{-}}$ is the determinant of the Cauchy matrix
$\\{1/(i^{+}_{j}+i^{-}_{k}-1)\\}_{j,k=1}^{n}$.

The fermions for the sinh-Gordon model are obtained from the CFT ones simply
by multiplication by a constant:

$\displaystyle\boldsymbol{\beta}^{\ast}_{2m-1}=D_{2m-1}(a)\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\qquad,\qquad\boldsymbol{\gamma}^{\ast}_{2m-1}=D_{2m-1}(Q-a)\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}\;,$
$\displaystyle\overline{\boldsymbol{\gamma}}^{\ast}_{2m-1}=D_{2m-1}(a)\overline{\boldsymbol{\gamma}}^{\textrm{CFT}\,\ast}_{2m-1}\qquad,\qquad\overline{\boldsymbol{\beta}}^{\ast}_{2m-1}=D_{2m-1}(Q-a)\overline{\boldsymbol{\beta}}^{\textrm{CFT}\,\ast}_{2m-1}\;,$

where

(2.22) $D_{2m-1}(a)=\frac{1}{2\pi
i}\left(\frac{\boldsymbol{\mu}\Gamma(1+b^{2})}{b^{1+b^{2}}}\right)^{-\frac{2m-1}{1+b^{2}}}\frac{\Gamma\left(\frac{a}{Q}+\frac{2m-1}{2bQ}\right)\Gamma\left(\frac{Q-a}{Q}+b\frac{2m-1}{2Q}\right)}{(m-1)!}\;.$

Note that this definition for the constants $D_{2m-1}$ differs from the one
used in Refs. [4]-[6] by the factor

(2.23)
$(-1)^{m}\sqrt{\frac{1+b^{2}}{i}}\frac{\boldsymbol{\mu}^{-\frac{2m-1}{1+b^{2}}}}{2\sin\left[\pi\left(\frac{a}{Q}-b\frac{2m-1}{2Q}\right)\right]}\;.$

The reason for this choice is twofold: on one side, the presence of
$\boldsymbol{\mu}^{-\frac{2m-1}{1+b^{2}}}$ makes the fermions dimensionless
while, on the other, the Q-periodic
$\sin\left[\pi\left(\frac{a}{Q}-b\frac{2m-1}{2Q}\right)\right]$ lets the non-
CFT fermions inherit the duality (2.19). Of course this last holds iff the
following term is “self-dual”

(2.24) $\frac{[\boldsymbol{\mu}\Gamma(1+b^{2})]^{\frac{1}{1+b^{2}}}}{b}\;,$

but this follows automatically when expressing $\boldsymbol{\mu}$ in terms of
the sinh-Gordon particle mass, which is explicitly self-dual:

(2.25)
$\boldsymbol{\mu}\Gamma(1+b^{2})=\left[\frac{m}{4\sqrt{\pi}}\Gamma\left(\frac{1}{2(1+b^{2})}\right)\Gamma\left(1+\frac{b^{2}}{2(1+b^{2})}\right)\right]^{1+b^{2}}\;.$

The constants $t_{\ell}(a)$ and $C_{m}(a)$ introduced in (1.7) are defined as
follows

(2.26)
$t_{\ell}(a)\doteq-\frac{1}{2}\sin^{-1}\left[\frac{\pi}{Q}\left(2a+\frac{\ell}{b}\right)\right]$

$\displaystyle C_{m}(a)\doteq\prod_{j=0}^{m-1}C_{1}(a-2bj)\;,$ $\displaystyle
C_{1}(a)\doteq[\boldsymbol{\mu}\Gamma(1+b^{2})]^{4x}\frac{\gamma(x)\gamma\left(\frac{1}{2}-x\right)}{2bQ\gamma(2bxQ)}\;,$

where $2Qx=2a-b$ and we denote $\gamma(y)\doteq\Gamma(y)/\Gamma(1-y)\;,\
\forall y\in\mathbb{C}$.

##  3\. TBA and one-point functions

As said above, since the TBA equations for the sinh-Gordon model are extremely
simple, it is quite straightforward to chose them as a starting point and
proceed to the construction of the function $\Theta(l,m|\alpha)$ relying on
the consistency equations which derive from the symmetries of the fermions.
Let us consider the sinh-Gordon model defined on an infinite cylinder of
circumference $2\pi R$; we call the infinite direction the _space direction_
and the compact one the _Matsubara direction_. The TBA for this model consists
of a single integral equation:

(3.28) $\epsilon(\theta)=2\pi
mR\cosh\theta-\int\limits_{-\infty}^{\infty}\Phi(\theta-\theta^{\prime})\log\left(1+e^{-\epsilon(\theta^{\prime})}\right)d\theta^{\prime}\;,$

with

$\displaystyle\Phi(\theta)$ $\displaystyle=\frac{1}{2\pi\cosh\left(\theta+\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)}+\frac{1}{2\pi\cosh\left(\theta-\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)}=\int\limits_{-\infty}^{\infty}e^{ik\theta}\widehat{\Phi}(k)\frac{dk}{2\pi}\;,$
$\displaystyle\widehat{\Phi}(k)$
$\displaystyle=\frac{\cosh\left(\pi\frac{b^{2}-1}{2(b^{2}+1)}k\right)}{\cosh\left(\pi\frac{k}{2}\right)}\;.$

Starting from this basic equation, one can build Baxter $Q$-functions in the
Matsubara direction; namely define

(3.30) $\log Q(\theta)=-\frac{\pi
mR\cosh\theta}{\sin\frac{\pi}{b^{2}+1}}+\int\limits_{-\infty}^{\infty}\frac{\log\left(1+e^{-\epsilon(\theta^{\prime})}\right)}{\cosh(\theta-\theta^{\prime})}\frac{d\theta^{\prime}}{2\pi}\;,$

where we have chosen the first term on the right-hand side for consistency. It
is straightforward to check that

(3.31) $e^{-\epsilon(\theta)}=Q\left(\theta+\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)Q\left(\theta-\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)\;,$

from which, recalling the Dirac delta representation
$\cosh(\theta+i\frac{\pi}{2})+\cosh(\theta-i\frac{\pi}{2})=2\pi\delta(\theta)$,
one can derive the bilinear equation555Actually, one should be careful and
define correctly the analyticity conditions for the function $Q(\theta)$; a
discussion can be found in Ref. [18].

(3.32) $Q\left(\theta+\frac{\pi i}{2}\right)Q\left(\theta-\frac{\pi
i}{2}\right)-Q\left(\theta+\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)Q\left(\theta-\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)=1\;.$

Introducing $\zeta=e^{(b^{2}+1)\theta}$, it’s not difficult to see how (3.32)
implies that the function $T(\zeta)$, defined from the equation

(3.33) $T(\zeta)Q(\theta)=Q\left(\theta+\pi
i\frac{b^{2}}{b^{2}+1}\right)+Q\left(\theta-\pi
i\frac{b^{2}}{b^{2}+1}\right)\;,$

is a single-valued function of $\zeta^{2}$, with essential singularities at
$\zeta=0$ and $\zeta=\infty$. This equation is a second order finite
difference equation for the function $Q(\theta)$ and thus admits two different
solutions: $Q(\theta)$ and $Q(\theta+i\frac{\pi}{b^{2}+1})$, the equation
(3.32) being their quantum Wronskian.

It’s important to stress that the equations for the functions $Q(\theta)$ and
$T(\theta)$ given here are to be considered as _definitions_ , so one should
check that they are reasonable. A verification of the correctness of these
definition was carried out in Ref. [15], where the behaviour of $T(\zeta)$ in
the ultraviolet region $R\rightarrow 0$ is investigated numerically, showing
how the asymptotics of $T(\zeta)$ for $\zeta\rightarrow 0$ and for
$\zeta\rightarrow\infty$ correcly reproduce the eigenvalues of CFT integrals
of motion and, moreover, that their normalisation is the same as in the sine-
Gordon case [19, 20, 21]; this is an extremely convincing argument.

Now, having the TBA equation (3.28) at our disposal, we introduce a deformed
kernel $\Phi_{\alpha}(\theta)$ requiring that its Fourier image
$\widehat{\Phi}(k,\alpha)$ satisfy $\widehat{\Phi}(k,0)=\widehat{\Phi}(k)$ and
the following symmetries

$\displaystyle\widehat{\Phi}(k,\alpha+$ $\displaystyle
2)=\widehat{\Phi}(k,\alpha)\;,\qquad\widehat{\Phi}(k,-\alpha)=\widehat{\Phi}(-k,\alpha)\;,$
$\displaystyle\widehat{\Phi}(k,\alpha-2\frac{b^{2}}{b^{2}+1})=\widehat{\Phi}(k+2i,\alpha)\;.$

The first two relations directly derive from the request that the fermions
transform in the correct way under the transformations $\sigma_{1}$ and
$\sigma_{2}$; the third one, on the other hand, is necessary in order to grant
the validity of the shift relation (1.7), as was shown in Ref. [8].

It’s not hard to find that the kernel we’re looking for has the following
form:

$\displaystyle\Phi_{\alpha}(\theta)=\frac{e^{i\pi\alpha}}{2\pi\cosh\left(\theta+\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)}+\frac{e^{-i\pi\alpha}}{2\pi\cosh\left(\theta-\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)}=\int\limits_{-\infty}^{\infty}e^{ik\theta}\widehat{\Phi}(k,\alpha)\frac{dk}{2\pi}\;,$
$\displaystyle\widehat{\Phi}(k,\alpha)=\frac{\cosh\left(\pi\frac{b^{2}-1}{2(b^{2}+1)}k-i\pi\alpha\right)}{\cosh\left(\pi\frac{k}{2}\right)}\;.$

It is interesting to notice that, contrary to the function
$\widehat{R}(k,\alpha)$ of the sine-Gordon model [6], the deformed kernel
$\widehat{\Phi}$ does not have poles in the $k$-plane whose positions depend
on $\alpha$. This simplification in the kernel structure is directly
correlated to the fact that sinh-Gordon one-point functions, as functions of
$\alpha$, have much simpler analytical properties than those of sine-Gordon.

Let us proceed by defining the dressed resolvent, which satisfies to the
equation

(3.36)
$R_{\textrm{dress}}(\theta,\theta^{\prime}|\alpha)-\left[\Phi\ast\,R_{\textrm{dress}}\right](\theta,\theta^{\prime}|\alpha)=\Phi(\theta,\theta^{\prime}|\alpha)\;,$

where
$\Phi(\theta,\theta^{\prime}|\alpha)\equiv\Phi_{\alpha}(\theta-\theta^{\prime})$
and the $\ast$ denotes a deformed convolution

(3.37)
$[f\ast\,g](\theta,\theta^{\prime})\doteq\int\limits_{-\infty}^{\infty}f(\theta,\phi)g(\phi,\theta^{\prime})dm(\phi)\;,\qquad
dm(\phi)\doteq\frac{d\phi}{1+e^{\epsilon(\phi)}}\;.$

Now, using the dressed resolvent, we build the function
$\Theta^{\textrm{shG}}_{R}$:

(3.38)
$R_{\textrm{dress}}(\theta,\theta^{\prime}|\alpha)-\Phi_{\alpha}(\theta-\theta^{\prime})=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\frac{dl}{2\pi}\frac{dm}{2\pi}\widehat{\Phi}(l,\alpha)\Theta^{\textrm{shG}}_{R}(l,m|\alpha)\widehat{\Phi}(m,-\alpha)e^{il\theta+im\theta^{\prime}}\;.$

Straightforward calculations show that the function
$\Theta^{\textrm{shG}}_{R}$ satisfies the following equation

(3.39)
$\Theta^{\textrm{shG}}_{R}(l,m|\alpha)-G(l+m)-\int\limits_{-\infty}^{\infty}G(l-k)\widehat{\Phi}(k,\alpha)\Theta^{\textrm{shG}}_{R}(k,m|\alpha)\frac{dk}{2\pi}=0\;,$

with the function $G(k)$ being the $k$-moment of the measure $dm(\theta)$

(3.40)
$G(k)\doteq\int\limits_{-\infty}^{\infty}e^{-ik\theta}\frac{d\theta}{1+e^{\epsilon(\theta)}}\;.$

A useful way to express the function $\Theta^{\textrm{shG}}_{R}$ is the
following

(3.41)
$\Theta^{\textrm{shG}}_{R}(il,im|\alpha)=e_{l}\ast\,e_{m}+e_{l}\ast\,R_{\textrm{dress}}^{(\alpha)}\ast\,e_{m}\;,$

where we have introduced the shorthand notation $e_{l}(\theta)\doteq
e^{l\,\theta}$.

Since, for the ground state, the function $\epsilon(\theta)$ is even, from the
symmetries of $\widehat{\Phi}(k,\alpha)$ one easily derives the following
relations:

(3.42)
$\Theta^{\textrm{shG}}_{R}(l,m|-\alpha)=\Theta^{\textrm{shG}}_{R}(m,l|\alpha)\;,\qquad\Theta^{\textrm{shG}}_{R}(l,m|\alpha+2)=\Theta^{\textrm{shG}}_{R}(l,m|\alpha)\;,$

$\displaystyle\Theta^{\textrm{shG}}_{R}(l,m|\alpha-2\frac{b^{2}}{b^{2}+1})$
$\displaystyle-\Theta^{\textrm{shG}}_{R}(l+2i,m-2i|\alpha)=$ (3.43)
$\displaystyle=\frac{\Theta^{\textrm{shG}}_{R}(l+2i,-i|\alpha)\Theta^{\textrm{shG}}_{R}(i,m-2i|\alpha)}{\pi
t_{1}(\frac{Q}{2}\alpha)-\Theta^{\textrm{shG}}_{R}(i,-i|\alpha)}\;.$

As has been said in the introduction, the function $\Theta_{R}^{\textrm{shG}}$
can be used in order to calculate the expectation values of descendants in the
fermionic basis:

######  Conjecture 1.

We conjecture that, as in the sine-Gordon model, the one-point functions of
the sinh-Gordon model in the fermionic basis are expressed in terms of a
determinant

(3.44)
$\frac{\langle\boldsymbol{\beta}^{\ast}_{I^{+}}\overline{\boldsymbol{\beta}}^{\ast}_{\overline{I}^{+}}\overline{\boldsymbol{\gamma}}^{\ast}_{\overline{I}^{-}}\boldsymbol{\gamma}^{\ast}_{I^{-}}\Phi_{\alpha}(0)\rangle_{R}}{\langle\Phi_{\alpha}(0)\rangle_{R}}=\mathcal{D}\left(I^{+}\cup(-\overline{I}^{+})|I^{-}\cup(-\overline{I}^{-})|\alpha\right)\;,$

where, for two sets $A=\\{a_{j}\\}_{j=1}^{n}$ and $B=\\{b_{j}\\}_{j=1}^{n}$,
we have

(3.45)
$\mathcal{D}(A|B|\alpha)\doteq\left(\prod_{\ell=1}^{n}\frac{\textrm{sgn}(a_{\ell})\textrm{sgn}(b_{\ell})}{\pi}\right)\det\left[\Theta^{\textrm{shG}}_{R}(ia_{j},ib_{k}|\alpha)-\pi\delta_{a_{j},-b_{k}}\textrm{sgn}(a_{j})t_{a_{j}}(\alpha)\right]_{j,k=1}^{n}$

Notice how, since
$\Theta^{\textrm{shG}}_{R}(l,m|\alpha)\underset{R\rightarrow\infty}{\rightarrow}0$,
in the infinite volume limit $R\rightarrow\infty$ the formulae for the one-
point functions in sinh-Gordon coincide with the analytic continuation with
respect to $b$ of the corresponding ones in sine-Gordon model [6].

##  4\. Numerical analysis in the $R\rightarrow 0$ limit

We now turn to the numerical evaluation of the one-point functions of the
sinh-Gordon model in the UV limit $R\rightarrow 0$. We will begin by studying
the behaviour of the descendant fields and then move to the primary ones. As
mentioned in the introduction, we rescale the model on a cylinder of radius
$2\pi$ and take $r=2\pi mR$ as the parameter to be sent to zero.

###  4.1. Descendant fields

We are interested in the UV behaviour of the following class of one-point
functions

(4.46)
$F_{2j-1,2k-1}(\alpha,r)\doteq\frac{\langle\boldsymbol{\beta}^{\ast}_{2j-1}\boldsymbol{\gamma}^{\ast}_{2k-1}\Phi_{\alpha}\rangle_{r}}{\langle\Phi_{\alpha}\rangle_{r}}\;,\qquad
j,k\in\mathbb{N}\;,$

which can be rewritten using (LABEL:eq:cfttosinhfermions) as

$F_{2j-1,2k-1}(\alpha,r)=D_{2j-1}(\alpha)D_{2k-1}(2-\alpha)\frac{\langle\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2j-1}\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2k-1}\Phi_{\alpha}(0)\rangle_{r}}{\langle\Phi_{\alpha}(0)\rangle_{r}}\;,$

In the $r\rightarrow 0$ limit, these functions should behave like ratios of
CFT one-point functions. In particular, using the formulae found in the
appendix of Ref. [6], we see that

(4.47) $F_{2j-1,2k-1}\underset{r\rightarrow 0}{\sim}-\left(\frac{2\pi
m}{r}\right)^{2j+2k-2}\frac{D_{2j-1}(\alpha)D_{2k-1}(2-\alpha)}{j+k-1}\Omega_{2j-1,2k-1}\;,$

where $\Omega_{2j-1,2k-1}$ are functions of the vacuum eigenvalues $I_{2n-1}$
of the integrals of motion, which can be found, for example, in Ref. [20]. For
the cases we are interested in we have

$\displaystyle\Omega_{1,1}(\alpha,r)=I_{1}(r)-\frac{\Delta_{\alpha}}{12}\;,$
$\displaystyle\Omega_{1,3}(\alpha,r)=I_{3}(r)-\frac{\Delta_{\alpha}}{6}I_{1}(r)+\frac{\Delta_{\alpha}^{2}}{144}+\frac{c+5}{1080}\Delta_{\alpha}-\frac{\Delta_{\alpha}}{360}d_{\alpha}\;,$

where

(4.49) $\Delta_{\alpha}=\frac{Q^{2}}{4}\alpha(2-\alpha)\;,\qquad
d_{\alpha}=\frac{1}{6}\sqrt{(25-c)(24\Delta_{\alpha}+1-c)}$

The vacuum eigenvalues of the integrals of motion do not depend directly on
the radius $r$, but rather on the momentum $P(r)$, which is itself a function
of $r$:

(4.50) $I_{1}(r)=P(r)^{2}-\frac{1}{24}\;,\qquad
I_{3}(r)=I_{1}(r)^{2}+\frac{1}{6}I_{1}(r)+\frac{c}{1440}\;.$

As explained neatly in Ref. [17], in the limit $r\rightarrow 0$ the main
contribution to the one-point functions $\langle e^{a\eta}\rangle$, with
$a>0$, comes from the following region in the configuration space

(4.51) $|b\eta_{0}|<-\log\frac{\mu^{2}}{\sin\pi b^{2}}\;,$

where $\eta_{0}$ is the zero mode of the field $\eta(z,\overline{z})$; here
the interaction term in sinh-Gordon action can be neglected. This means that
in this region we can consider $\eta$ as a free field and that the ground
state wave functional $\boldsymbol{\Psi}_{0}[\eta]$ can be approximated by the
superposition of two zero-mode plane waves

(4.52)
$\boldsymbol{\Psi}_{0}[\eta]\underset{r\rightarrow\infty}{\sim}\left(c_{1}e^{iP(r)\eta_{0}}+c_{2}e^{-iP(r)\eta_{0}}\right)\;,$

where the momentum $P(r)$ is quantised thanks to the presence of the potential
walls $b\eta_{0}\sim\pm\log\frac{\mu^{2}}{\sin\pi b^{2}}$. The quantisation
condition reads

(4.53) $S(P)^{2}=1\;\Rightarrow\;\delta(P)=\pi\;,\ S(P)\doteq
e^{-i\delta(P)}\;,$

where $S(P)$ is the Liouville reflection amplitude

(4.54)
$S(P)=-\left(\boldsymbol{\mu}\frac{\Gamma(1+b^{2})}{b^{2}}\right)^{-4i\frac{P(r)}{b}}\frac{\Gamma\big{(}1+2iP(r)b\big{)}\Gamma\big{(}1+2iP(r)b^{-1}\big{)}}{\Gamma\big{(}1-2iP(r)b\big{)}\Gamma\big{(}1-2iP(r)b^{-1}\big{)}}\;.$

Using (1.8) and remembering that we rescaled the mass $m\rightarrow mR$, we
easily obtain the quantisation condition for the momentum

$\displaystyle
2P(r)Q\log\left[\frac{r}{8\pi^{\frac{3}{2}}\big{(}b^{2}\big{)}^{\frac{1}{1+b^{2}}}}\Gamma\Big{(}\frac{1}{2(1+b^{2})}\Big{)}\Gamma\Big{(}1+\frac{b^{2}}{2(1+b^{2})}\Big{)}\right]=$
$\displaystyle=-\frac{\pi}{2}+\frac{1}{2i}\log\left[\frac{\Gamma\big{(}1+2iP(r)b\big{)}\Gamma\big{(}1+2iP(r)b^{-1}\big{)}}{\Gamma\big{(}1-2iP(r)b\big{)}\Gamma\big{(}1-2iP(r)b^{-1}\big{)}}\right]\;.$

We have considered the two following ratios of expectation values

(4.56)
$F_{1,1}(\alpha,r)\doteq\frac{\langle\boldsymbol{\beta}^{\ast}_{1}\boldsymbol{\gamma}^{\ast}_{1}\Phi_{\alpha}(0)\rangle_{r}}{\langle\Phi_{\alpha}(0)\rangle_{r}}\;,\qquad
F_{1,3}(\alpha,r)\doteq\frac{\langle\boldsymbol{\beta}^{\ast}_{1}\boldsymbol{\gamma}^{\ast}_{3}\Phi_{\alpha}(0)\rangle_{r}}{\langle\Phi_{\alpha}(0)\rangle_{r}}\;,$

and evaluated numerically the corresponding functions
$\Theta_{r}^{\textrm{shG}}(i,i|\alpha)$ and
$\Theta_{r}^{\textrm{shG}}(i,3i|\alpha)$ for values of $\alpha$ ranging from
$0.75$ up to $1.5$, with $b\in[0.4,1.0]$ and $r\in[0.005,0.95]$. Figures 1-8
show some of these numerical estimates plotted against the curve (4.47); the
agreement of the data with the theoretical prevision is very good for the
whole range of $r$ considered.

Figure 1. Plot of $F_{1,1}(\alpha,r)$ against its theoretical behaviour for
$\alpha=0.75$ and $b=0.4$

Figure 2. Plot of $F_{1,1}(\alpha,r)$ against its theoretical behaviour for
$\alpha=0.75$ and $b=0.8$

Figure 3. Plot of $F_{1,1}(\alpha,r)$ against its theoretical behaviour for
$\alpha=1.1$ and $b=0.4$

Figure 4. Plot of $F_{1,1}(\alpha,r)$ against its theoretical behaviour for
$\alpha=1.1$ and $b=0.8$

Figure 5. Plot of $F_{1,3}(\alpha,r)$ against its theoretical behaviour for
$\alpha=0.75$ and $b=0.4$

Figure 6. Plot of $F_{1,3}(\alpha,r)$ against its theoretical behaviour for
$\alpha=0.75$ and $b=0.8$

Figure 7. Plot of $F_{1,3}(\alpha,r)$ against its theoretical behaviour for
$\alpha=1$ and $b=0.4$

Figure 8. Plot of $F_{1,3}(\alpha,r)$ against its theoretical behaviour for
$\alpha=1$ and $b=0.8$

The tables 1 and Tab.2, displaying the values of the relative error $\sigma$

(4.57)
$\sigma_{2j-1,2k-1}\doteq\left|1-\frac{F_{2j-1,2k-1}(\alpha,r)}{F_{2j-1,2k-1}^{\textrm{CFT}}(\alpha,r)}\right|$

with

(4.58) $F_{2j-1,2k-1}^{\textrm{CFT}}(\alpha,r)=-\left(\frac{2\pi
m}{r}\right)^{2j+2k-2}\frac{D_{2j-1}(\alpha)D_{2k-1}(2-\alpha)}{j+k-1}\Omega_{2j-1,2k-1}\;,$

are a remarkable evidence in support of the conjecture 3.45.

Table 1. Values of the relative error for $F_{1,1}(\alpha,r)$. | $\sigma_{1,1}$
---|---
| $\alpha=0.75$ |  | $\alpha=1.1$
$r$ | $b=0.4$ | $b=0.8$ |  | $b=0.4$ | $b=0.8$
0.005 | $1.5\times 10^{-4}$ | $2.0\times 10^{-5}$ |  | $2.4\times 10^{-4}$ | $6.0\times 10^{-5}$
0.01 | $5.5\times 10^{-5}$ | $3.3\times 10^{-6}$ |  | $1.1\times 10^{-4}$ | $1.5\times 10^{-5}$
0.015 | $2.3\times 10^{-5}$ | $1.2\times 10^{-6}$ |  | $6.1\times 10^{-5}$ | $8.1\times 10^{-6}$
0.02 | $1.3\times 10^{-5}$ | $1.8\times 10^{-6}$ |  | $3.7\times 10^{-5}$ | $4.6\times 10^{-6}$
0.025 | $7.5\times 10^{-6}$ | $3.0\times 10^{-7}$ |  | $2.2\times 10^{-5}$ | $2.3\times 10^{-6}$
0.03 | $5.1\times 10^{-6}$ | $7.3\times 10^{-7}$ |  | $1.6\times 10^{-5}$ | $2.8\times 10^{-6}$
0.035 | $1.7\times 10^{-6}$ | $1.1\times 10^{-6}$ |  | $1.1\times 10^{-5}$ | $1.1\times 10^{-6}$
0.04 | $1.4\times 10^{-6}$ | $1.1\times 10^{-6}$ |  | $7.0\times 10^{-6}$ | $3.1\times 10^{-7}$
0.045 | $1.4\times 10^{-6}$ | $1.1\times 10^{-6}$ |  | $6.7\times 10^{-6}$ | $2.2\times 10^{-6}$
0.05 | $1.3\times 10^{-6}$ | $1.3\times 10^{-6}$ |  | $2.4\times 10^{-6}$ | $2.5\times 10^{-6}$
0.055 | $4.2\times 10^{-6}$ | $3.3\times 10^{-6}$ |  | $7.1\times 10^{-6}$ | $8.0\times 10^{-7}$
0.06 | $1.1\times 10^{-6}$ | $2.5\times 10^{-6}$ |  | $2.2\times 10^{-6}$ | $3.2\times 10^{-7}$
0.065 | $4.0\times 10^{-7}$ | $2.4\times 10^{-7}$ |  | $2.4\times 10^{-6}$ | $4.4\times 10^{-7}$
0.07 | $2.7\times 10^{-7}$ | $2.9\times 10^{-7}$ |  | $2.2\times 10^{-6}$ | $1.7\times 10^{-7}$
0.075 | $3.1\times 10^{-7}$ | $2.8\times 10^{-7}$ |  | $1.3\times 10^{-6}$ | $1.2\times 10^{-6}$
0.08 | $1.3\times 10^{-7}$ | $1.0\times 10^{-7}$ |  | $1.0\times 10^{-6}$ | $4.3\times 10^{-8}$
0.085 | $5.4\times 10^{-7}$ | $1.3\times 10^{-7}$ |  | $3.1\times 10^{-8}$ | $5.4\times 10^{-7}$
0.09 | $2.8\times 10^{-8}$ | $2.2\times 10^{-7}$ |  | $1.4\times 10^{-6}$ | $2.8\times 10^{-6}$
0.095 | $1.2\times 10^{-6}$ | $2.3\times 10^{-7}$ |  | $2.6\times 10^{-6}$ | $6.8\times 10^{-11}$
Table 2. Values of the relative error for $F_{1,3}(\alpha,r)$. | $\sigma_{1,3}$
---|---
| $\alpha=0.75$ |  | $\alpha=1$
$r$ | $b=0.4$ | $b=0.8$ |  | $b=0.4$ | $b=0.8$
0.005 | $2.4\times 10^{-4}$ | $6.2\times 10^{-5}$ |  | $1.4\times 10^{-4}$ | $1.8\times 10^{-5}$
0.01 | $1.0\times 10^{-4}$ | $1.6\times 10^{-5}$ |  | $5.2\times 10^{-5}$ | $5.0\times 10^{-6}$
0.015 | $6.2\times 10^{-5}$ | $9.3\times 10^{-6}$ |  | $2.5\times 10^{-5}$ | $3.1\times 10^{-6}$
0.02 | $3.7\times 10^{-5}$ | $4.1\times 10^{-6}$ |  | $1.2\times 10^{-5}$ | $1.1\times 10^{-6}$
0.025 | $2.1\times 10^{-5}$ | $8.9\times 10^{-7}$ |  | $7.0\times 10^{-6}$ | $5.0\times 10^{-7}$
0.03 | $1.8\times 10^{-5}$ | $8.1\times 10^{-7}$ |  | $4.2\times 10^{-6}$ | $8.8\times 10^{-7}$
0.035 | $1.0\times 10^{-5}$ | $8.5\times 10^{-7}$ |  | $1.7\times 10^{-6}$ | $9.9\times 10^{-7}$
0.04 | $7.4\times 10^{-6}$ | $1.4\times 10^{-6}$ |  | $1.0\times 10^{-6}$ | $6.7\times 10^{-7}$
0.045 | $5.9\times 10^{-6}$ | $1.3\times 10^{-6}$ |  | $4.1\times 10^{-7}$ | $2.1\times 10^{-6}$
0.05 | $3.3\times 10^{-6}$ | $2.2\times 10^{-6}$ |  | $3.4\times 10^{-7}$ | $1.0\times 10^{-6}$
0.055 | $2.4\times 10^{-6}$ | $1.2\times 10^{-6}$ |  | $1.3\times 10^{-6}$ | $2.2\times 10^{-6}$
0.06 | $2.2\times 10^{-6}$ | $4.8\times 10^{-7}$ |  | $7.1\times 10^{-7}$ | $8.0\times 10^{-7}$
0.065 | $1.6\times 10^{-6}$ | $7.0\times 10^{-7}$ |  | $3.9\times 10^{-7}$ | $4.2\times 10^{-8}$
0.07 | $1.3\times 10^{-8}$ | $4.9\times 10^{-7}$ |  | $3.0\times 10^{-7}$ | $4.7\times 10^{-7}$
0.075 | $3.8\times 10^{-7}$ | $1.1\times 10^{-6}$ |  | $7.6\times 10^{-9}$ | $5.8\times 10^{-7}$
0.08 | $8.0\times 10^{-7}$ | $1.2\times 10^{-6}$ |  | $9.2\times 10^{-8}$ | $3.6\times 10^{-7}$
0.085 | $2.8\times 10^{-6}$ | $4.2\times 10^{-7}$ |  | $4.5\times 10^{-7}$ | $6.1\times 10^{-7}$
0.09 | $1.6\times 10^{-6}$ | $2.8\times 10^{-6}$ |  | $7.0\times 10^{-7}$ | $4.0\times 10^{-7}$
0.095 | $3.0\times 10^{-6}$ | $1.0\times 10^{-6}$ |  | $7.1\times 10^{-7}$ | $6.1\times 10^{-7}$

###  4.2. Primary fields

Let us now consider the following ratio of primary fields’ expectation values

(4.59)
$\mathcal{F}(\alpha,r)\doteq\frac{\langle\Phi_{\alpha-2\frac{b^{2}}{b^{2}+1}}\rangle^{\textrm{shG}}_{r}}{\langle\Phi_{\alpha}\rangle^{\textrm{shG}}_{r}}\;.$

Using the shift formula (1.7) and the determinant one (3.44) we can write

(4.60)
$\mathcal{F}(\alpha,r)=\frac{C_{1}(\alpha)}{t_{1}(\alpha)}\frac{\langle\boldsymbol{\beta}^{\ast}_{1}\bar{\boldsymbol{\gamma}}^{\ast}_{1}\Phi_{\alpha}\rangle^{\textrm{shG}}_{r}}{\langle\Phi_{\alpha}\rangle^{\textrm{shG}}_{r}}=-\frac{C_{1}(\alpha)}{\pi
t_{1}(\alpha)}\Big{[}\Theta(i,-i|\alpha)-\pi t_{1}(\alpha)\Big{]}\;.$

On the other hand, from Ref. [15] we know that we can approximate the
behaviour of the expectation value of a primary field $\Phi_{\alpha}$ in the
region (4.51) with that of a three-point function of Liouville CFT:

(4.61) $\langle\Phi_{\alpha}\rangle^{\textrm{shG}}_{r}\underset{r\rightarrow
0}{\sim}\mathcal{N}(r,b)\langle
0|e^{a(-P)\eta(-\infty)}\Phi_{\alpha}e^{a(P)\eta(\infty)}|0\rangle_{r}^{\textrm{Liou}}$

where the function $\mathcal{N}(r,b)$ is a normalization constant and

(4.62)
$a(P)=\frac{Q}{2}+iP(r)\;\Rightarrow\qquad\Delta_{a(P)}=\frac{Q^{2}}{4}-P(r)^{2}$

with $P(r)$ satisfying the quantization condition (LABEL:eq:quantcond).

The form of Liouville three-point function was found in Refs. [16] and [17]
and reads

(4.63) $\langle
0|e^{a(-P)\eta(-\infty)}\Phi_{\alpha}e^{a(P)\eta(\infty)}|0\rangle_{r}^{\textrm{Liou}}=\left(\boldsymbol{\mu}\frac{\Gamma(1+b^{2})}{b^{1+b^{2}}}\right)^{-Q\frac{\alpha}{b}}\Upsilon_{0}\frac{\Upsilon(2a)\Upsilon(Q-2iP)\Upsilon(Q+2iP)}{\Upsilon(a)^{2}\Upsilon(a-2iP)\Upsilon(a+2iP)}\;,$

where the function $\Upsilon(x)$ is defined by the equations

$\frac{\Upsilon(x+b)}{\Upsilon(x)}=\gamma(b\,x)b^{1-2bx}\;,\quad\frac{\Upsilon(x+b^{-1})}{\Upsilon(x)}=\gamma\left(\frac{x}{b}\right)b^{-1+2\frac{x}{b}}\;,\quad\Upsilon_{0}\doteq\frac{d\Upsilon}{dx}\Big{|}_{x=0}\;.$

The general form of the normalization $\mathcal{N}(r,b)$ is not known, but
this is irrelevant to our needs, since we are considering the ratio of two
one-point functions.

With some simple calculations one finds

$\displaystyle\mathcal{F}(\alpha,r)$ $\displaystyle\underset{r\rightarrow
0}{\sim}\mathcal{F}^{\textrm{CFT}}(\alpha,r)=\left[\frac{r}{8\pi^{\frac{3}{2}}}\Gamma\left(\frac{1}{2(1+b^{2})}\right)\Gamma\left(1+\frac{b^{2}}{2(1+b^{2})}\right)\right]^{2}\times$
(4.64)
$\displaystyle\times\frac{\gamma\big{(}b(a-b)\big{)}^{2}}{\gamma\big{(}b(2a-b)\big{)}\gamma\big{(}2b(a-b)\big{)}}\gamma\big{(}b(a-b+2iP)\big{)}\gamma\big{(}b(a-b-2iP)\big{)}\;.$

We have evaluated numerically the function $\Theta(i,-i|\alpha)$ and used it
to extract the value of $\mathcal{F}(\alpha,r)$ by means of the formula
(4.60). We then compared the data we obtained with the theoretical CFT
behaviour (4.64). Figures 9-13 show the results for various values of $\alpha$
and $b$.

Figure 9. Plot of $\mathcal{F}(\alpha,r)$ against its theoretical behaviour
for $\alpha=0.75$ and $b=0.4$

Figure 10. Plot of $\mathcal{F}(\alpha,r)$ against its theoretical behaviour
for $\alpha=0.75$ and $b=0.7$

Figure 11. Plot of $\mathcal{F}(\alpha,r)$ against its theoretical behaviour
for $\alpha=0.75$ and $b=0.8$

Figure 12. Plot of $\mathcal{F}(\alpha,r)$ against its theoretical behaviour
for $\alpha=1.5$ and $b=0.4$

Figure 13. Plot of $\mathcal{F}(\alpha,r)$ against its theoretical behaviour
for $\alpha=1.5$ and $b=0.8$

In table 3 are collected the values of the relative error $\varsigma$

(4.65)
$\varsigma\doteq\left|1-\frac{\mathcal{F}(\alpha,r)}{\mathcal{F}^{\textrm{CFT}}(\alpha,r)}\right|\;.$

Table 3. Values of the relative error for $\mathcal{F}(\alpha,r)$. | $\varsigma$
---|---
| $\alpha=0.75$ |  | $\alpha=1.5$
$r$ | $b=0.4$ | $b=0.7$ | $b=0.8$ |  | $b=0.4$ | $b=0.8$
0.005 | $6.2\times 10^{-3}$ | $1.1\times 10^{-3}$ | $1.2$ |  | $1.1\times 10^{-2}$ | $8.2\times 10^{-4}$
0.01 | $2.7\times 10^{-3}$ | $1.7\times 10^{-3}$ | $1.2$ |  | $3.6\times 10^{-3}$ | $2.5\times 10^{-4}$
0.015 | $1.4\times 10^{-3}$ | $5.9\times 10^{-3}$ | $1.3$ |  | $1.7\times 10^{-3}$ | $1.1\times 10^{-4}$
0.02 | $7.8\times 10^{-4}$ | $7.3\times 10^{-3}$ | $1.3$ |  | $9.6\times 10^{-4}$ | $5.4\times 10^{-5}$
0.025 | $5.7\times 10^{-4}$ | $9.4\times 10^{-3}$ | $1.3$ |  | $5.8\times 10^{-4}$ | $2.7\times 10^{-5}$
0.03 | $3.1\times 10^{-4}$ | $1.1\times 10^{-2}$ | $1.3$ |  | $3.7\times 10^{-4}$ | $1.6\times 10^{-5}$
0.035 | $2.2\times 10^{-4}$ | $1.2\times 10^{-2}$ | $1.3$ |  | $2.4\times 10^{-4}$ | $1.0\times 10^{-5}$
0.04 | $1.7\times 10^{-4}$ | $1.4\times 10^{-2}$ | $1.4$ |  | $1.7\times 10^{-4}$ | $5.3\times 10^{-6}$
0.045 | $1.6\times 10^{-4}$ | $1.6\times 10^{-2}$ | $1.4$ |  | $1.2\times 10^{-4}$ | $1.2\times 10^{-6}$
0.05 | $3.6\times 10^{-5}$ | $1.7\times 10^{-2}$ | $1.4$ |  | $8.5\times 10^{-5}$ | $5.9\times 10^{-6}$
0.055 | $9.5\times 10^{-5}$ | $1.9\times 10^{-2}$ | $1.4$ |  | $5.9\times 10^{-5}$ | $1.1\times 10^{-6}$
0.06 | $3.6\times 10^{-5}$ | $2.0\times 10^{-2}$ | $1.4$ |  | $4.4\times 10^{-5}$ | $6.6\times 10^{-7}$
0.065 | $7.2\times 10^{-5}$ | $2.1\times 10^{-2}$ | $1.4$ |  | $3.5\times 10^{-5}$ | $5.6\times 10^{-7}$
0.07 | $5.5\times 10^{-5}$ | $2.3\times 10^{-2}$ | $1.4$ |  | $2.6\times 10^{-5}$ | $9.8\times 10^{-7}$
0.075 | $2.8\times 10^{-5}$ | $2.4\times 10^{-2}$ | $1.4$ |  | $1.8\times 10^{-5}$ | $3.3\times 10^{-7}$
0.08 | $3.1\times 10^{-5}$ | $2.5\times 10^{-2}$ | $1.5$ |  | $1.3\times 10^{-5}$ | $9.2\times 10^{-8}$
0.085 | $3.1\times 10^{-5}$ | $2.7\times 10^{-2}$ | $1.5$ |  | $9.3\times 10^{-6}$ | $8.2\times 10^{-8}$
0.09 | $7.5\times 10^{-6}$ | $2.8\times 10^{-2}$ | $1.5$ |  | $9.1\times 10^{-6}$ | $2.0\times 10^{-7}$
0.095 | $3.7\times 10^{-6}$ | $2.9\times 10^{-2}$ | $1.5$ |  | $4.8\times 10^{-6}$ | $3.9\times 10^{-7}$

The agreement between the data and the CFT behaviour is incredibly good until
$b\gtrsim 0.7$, when $\alpha=0.75$, as is clearly visible from figures 10 and
11. The reason for this discrepancy is that, as we explained in the
introduction, the supposition that sinh-Gordon approaches naïvely the
Liouville CFT in its UV limit is no longer valid when
$b\geq\sqrt{\frac{\alpha}{2-\alpha}}$. When $\alpha=0.75$, the _critical_
value is $b^{\textrm{crit}}=\sqrt{3/5}\sim 0.774$, which explains why figure
10 still shows a good agreement for very small values of $r$, while in figure
11 we see that the data and the CFT curve behave in radically different ways.

##  5\. Conclusion

We investigated numerically the behaviour of the conjecture 3.45 in the UV
limit $R\rightarrow 0$ of the sinh-Gordon model defined on an infinite
cylinder of radius $2\pi R$. We found an extremely good agreement with the
theoretical predictions in Ref. [15], up to the $4^{\textrm{th}}$ decimal
place, in the cases of both primary and descendant fields. In figures 1-13 and
tables 1-3 part of these results are collected. We consider these, along with
the analytical results of Ref. [8], as a very strong confirmation of the
correctness of the fermionic basis description for the sinh-Gordon model.

We have also verified that the limiting behaviour of the primary fields’
expectation values is very well described by that of a particular three point
function in Liouville CFT only if the parameters are such that the scaling
dimensions of the involved fields are all positive, meaning that $0<b<a<Q$. It
would be interesting to study the behaviour of sinh-Gordon model’s UV limit
outside this region.

## Acknowledgments

I am grateful to F. Smirnov whose valuable advices helped considerably to
direct my analysis and organize this work.
Research of SN is supported by the People Programme (Marie Curie Actions) of
the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA
Grant Agreement No 317089 (GATIS).
This project was partially supported by INFN grant IS FTECP and the UniTo-
SanPaolo research grant Nr TO-Call3-2012-0088.

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