Neural text detectors are models trained to detect whether a given text was generated by a language model or written by a human. In this paper, we investigate three simple and resource-efficient strategies (parameter tweaking, prompt engineering, and character-level mutations) to alter texts generated by GPT-3.5 that are unsuspicious or unnoticeable for humans but cause misclassification by neural text detectors. The results show that especially parameter tweaking and character-level mutations are effective strategies.

§ INTRODUCTION

The widespread availability of neural text generation models, like ChatGPT, has caused an increased desire for neural text detectors, i.e. models that can detect whether a given text was AI-generated. The reliance of, e.g., educational institutions on such detectors has raised questions about their robustness in general and in specific with regard to adversarial attacks <cit.>. Such attacks exploit the fact that machine learning models by identifying patterns in the data rather than by understanding actual underlying concepts. Consequently, introducing small, human-unnoticeable perturbations can result in misclassification. <cit.>

Adversarial attacks can be categorised into black-box and white-box attacks <cit.>. In white-box attacks, the attacker has full access to the target model, including its parameters, architecture, and loss function <cit.>. During black-box attacks, the adversary can only input queries and observe the outputs without any insights into internal processing <cit.>. Furthermore, it can be distinguished between targeted and untargeted attacks, where targeted attacks aim at triggering misclassification towards a specific label, while untargeted aim to cause any misclassification <cit.>.

This paper investigates effective and resource-efficient universal attack strategies in a black-box scenario with minimal resources, based on text generated with GPT 3.5 and three neural text detectors: the widely used open source GPT-2 Output detector model[<https://github.com/openai/gpt-2-output-dataset/tree/master/detector>], the OpenAI text classifier[<https://platform.openai.com/ai-text-classifier>], and the commercial Turnitin AI detector[<https://www.turnitin.com>], which is used by many educational institutions. The results show that character-level mutations, tweaking the parameters of the generative model, as well as prompt engineering, are efficient and effective strategies, showing that currently available neural text detectors can not reliably detect texts generated by state-of-the-art large language models (LLMs).

§ RELATED WORK

Most of the existing literature about adversarial attacks focuses on image detection. Textual input is less used due to its discrete nature and the difficulty in introducing human-imperceptible perturbations, contrary to the image data, where a change in a few hundred pixels can go unnoticed  <cit.>. Examples of adversarial attacks on general text classification models include the work by <cit.> and <cit.>. More recent work has started to specifically look into adversarial attacks on neural text detectors: <cit.> showed that introducing spelling mistakes and replacing characters with homoglyphs can significantly reduce the detection rate for GPT-2 texts. <cit.> showed that similar character-level mutation-based attacks are also successful for RoBERTa-based detection models. <cit.> not only showed that existing detectors are vulnerable to simple rephrasing, but they also showed that they are biased towards flagging texts that have been (manually) written by non-native speakers as AI-generated.

Because currently available methods are vulnerable to adversarial attacks, multiple suggestions have been made to improve their robustness, e.g. by <cit.>, <cit.>, <cit.>, and <cit.>. While watermarking techniques to identify AI-generated texts are also investigated, they are generally seen as vulnerable to adversarial attacks, especially to mutation and paraphrasing-based approaches <cit.>.

§ APPROACHES

Based on the existing literature, we identified three promising and efficient approaches for adversarial attacks: parameter tweaking, prompt engineering and character-level mutations. All approaches were tested with the three neural text detectors mentioned in Section <ref>: GPT-2 output detector, OpenAI classifier, and Turnitin AI writing detector. The basis for all attacks were texts generated by the GPT-3.5-turbo model via the OpenAI API. The text samples were produced as 500-word essays, with topics taken from a list of 200 essay topics <cit.> and the prompt “Write a five-hundred-word argumentative essay on the topic ‘topic’.”. It was then evaluated how the detection rate changed between the original texts and their altered version. To ensure comparability of the results between different detectors, all scores were projected onto a scale from 0.0 (very likely not AI-generated) to 1.0 (very likely AI-generated). The GPT-2 Output detector returns a score between 0.0 and 1.0, that can be used directly. Turnitin returns a percentage between 0 and 100 indicating how much of the text was generated by AI. We divide the score by 100. The OpenAI classifier returns one of five labels (“very unlikely”, “unlikely”, “unclear”, “possibly”, “likely”). For each of the labels, <cit.> provides a corresponding range of numerical thresholds, of which we take the mean score (0.05, 0.275, 0.675, 0.94, 0.99). The code for the evaluation was written with the assistance of GPT-4, followed by extensive testing of the code, as well as additional, manually implemented, features. The code and the data are available on GitHub[<https://github.com/Lolya-cloud/adversarial-attacks-on-neural-text-detectors>].

Parameter     Min     Max     Default
Temperature     0.0     2.0     1.0

Top p     0.0     1.0     1.0

Frequency penalty     -2.0     2.0     0.0

Presence penalty    -2.0     2.0     0.0

Investigated parameters

ylabel = detection score,
legend style=nodes=scale=0.8, transform shape,
legend pos=north east]
+[mark=square] table [x=Frequency penalty, y=Turnitin, col sep=comma] data/frequency_penalty_combined_scores.csv;
+[mark=o] table [x=Frequency penalty, y=OpenAI classifier, col sep=comma] data/frequency_penalty_combined_scores.csv;
+[mark=triangle] table [x=Frequency penalty, y=GPT2 detector, col sep=comma] data/frequency_penalty_combined_scores.csv;
GPT2 detector

frequency penalty

legend pos=north east]
+[mark=square] table [x=Presence penalty, y=Turnitin, col sep=comma] data/presence_penalty_combined_scores.csv;
+[mark=o] table [x=Presence penalty, y=OpenAI classifier, col sep=comma] data/presence_penalty_combined_scores.csv;
+[mark=triangle] table [x=Presence penalty, y=GPT2 detector, col sep=comma] data/presence_penalty_combined_scores.csv;

presence penalty

Influence of the frequency and presence penalty on the detection score

[width=, xlabel=frequency penalty, ylabel=presence penalty]
[opacity=1, mark size=3pt, scatter, only marks] table [x index=0, y index=1, scatter src=3, col sep=comma] data/dataframe.csv;

GPT-2 detector

[width=, xlabel=frequency penalty]
[opacity=1, mark size=3pt, scatter, only marks] table [x index=0, y index=1, scatter src=4, col sep=comma] data/dataframe.csv;

OpenAI classifier

[width=, xlabel=frequency penalty, colorbar]
[opacity=1, mark size=3pt, scatter, only marks] table [x index=0, y index=1, scatter src=5, col sep=comma] data/dataframe.csv;

Influence of joined optimsation of fequency and presence penalties on the detection score

§.§ Parameter tweaking

First, we investigated the influence of GPT-3.5 generation parameters on the detection. Table <ref> shows the parameters we focus on because they have the biggest impact on the produced texts according to <cit.>. Temperature and top p control randomness in the text. By increasing the temperature, the output becomes more random. However, for values beyond the default of 1.0, the length of the outputs started fluctuating strongly and the quality of the texts dropped. Consequently, we focused on the range between 0.0 and 1.0. Top p represents the percentage of tokens selected based on their probability mass. The frequency penalty controls the frequency of tokens appearing in the text, with higher values leading to more diverse verbatim. During the testing phase, it was found that increasing the frequency penalty beyond 1.0 degrades the quality of the texts rapidly. Additionally, as decreasing the value below 0.0 increases the repetitiveness, we focused on the range between 0.0 and 1.0. Finally, the presence penalty controls the model’s likelihood of repeating tokens in the text. Higher values of presence penalty lead to the model producing more diverse texts. Following consideration similar to the frequency penalty, the negative values were discarded, and we focused on the range between 0.0 and 2.0. <cit.> First, we investigated each parameter separately, changing it in steps of 0.1. Subsequently, we used the two parameters that had the biggest impact and investigated whether their interaction also influences the detection rate by performing a grid search in steps of 0.1 with these parameters.

§.§ Prompt engineering

The second approach explored the effect of prompt engineering on detection rates. Since <cit.> have shown that detection is vulnerable towards simple rephrasing, our hypothesis was that providing regeneration instructions as a separate prompt might lower the detection rate. This hypothesis was tested with the following four prompts, each of which was used to generate ten texts: (1) standard prompt as described in Section; (2)
standard prompt followed by second query "Regenerate the essay."; (3) advanced prompt as shown in Appendix <ref>; (4) standard prompt followed by advanced prompt as second query. Additionally, several prompts to increase the perplexity and burstiness of generated texts were tested, a strategy that is popular among users (see e.g. <cit.>). Ten prompts were designed, and for each, ten texts were generated. The first five prompts use single-query architecture, while the others utilise a two-query concept. The following methods were tested with different prompts: (1) Explaining burstiness and perplexity, then asking to implement them (see Appendix <ref>); (2) Explicitly asking to maximise either perplexity, burstiness or both (standard prompt followed by “Maximize the burstiness / perplexity / burstiness and perplexity of the text.”); (3) Explicitly asking to rewrite to avoid detection. (“Rewrite the above essay in order to avoid AI detection.”)

§.§ Character-level mutations

This approach aimed to test the robustness of the detectors against traditional adversarial attack vectors. Three character level mutations were taken as a basis: replacing either Latin lowercase “a” or “e” with the corresponding Cyrillic analogue; replacing Latin lowercase “L” with Latin uppercase “I”. Ten texts were generated with the standard prompt and then the mutations were applied.

§ RESULTS

§.§ Parameter tweaking

The detection rate of all three detectors dropped with an increment in either frequency or presence penalty (see Figure <ref>). Starting from a frequency penalty of 0.3-0.4 and a presence penalty of 1.0-1.2, the detection rate fell under 50% with some fluctuations. An analysis of the generated texts showed that increasing either the frequency or presence penalty led to more diverse texts. A higher frequency penalty caused a wider vocabulary variety. For values above 0.6, the occurrence of punctuation mistakes and unclear wordings quickly increased, making the texts difficult to read. For values between 0.0 and 0.6, increases in value caused an incremental increase in text complexity while preserving quality and readability. A higher presence penalty primarily influenced the diversity of perspectives and text engagement. However, for values above 0.6, the coherence and logical progression rapidly decreased and texts became less subject-focused. Therefore, increasing either frequency or presence penalty from the default of 0.0 up to 0.6 can be seen as successful attack strategies that significantly lowers detection while maintaining text quality. Increasing the temperature and top p value above the default was already excluded from the experimental design, because of the strong negative effects on text quality. Lowering either value resulted in more deterministic outputs leading to higher detection rates, therefore, tweaking those parameters is not a successful attack strategy.

In a second step, the interaction between frequency and presence penalty was investigated. Figure <ref> shows that the detection rates dropped for all three detectors with an increment in frequency and presence penalty. Notably, they started dropping for smaller values of presence and frequency penalty than they did for the individual parameters, thereby minimising the potential negative effects on text quality. The GPT-2 detector showed the worst performance in the comparison with very quickly declining detection scores.

§.§ Prompt engineering

The simple regeneration approach, whether using a second query or providing detailed instruction in the first query, did not have an impact on detection rates. Increasing the perplexity and burstiness of the texts through prompts, on the other hand, caused a drop in the detection rate across all three detectors, however only if applied in two separate prompts, as shown in Figure <ref>. Although the approach managed to decrease the detection rate across all three detectors, the score sank only for the GPT-2 detector below 0.5 (see Figure <ref>). For the other two detectors, the score stayed above 0.5, meaning that the generated texts would still be detected as AI-generated or at least as undecidable.

[col sep=comma]data/gpt2_scores.csv
boxplot/draw direction = y,
xtick = 1, 2, 3, 4,5,6,7,8,9,10,
xticklabel style = align=center, font=, rotate=30,
xticklabels = std (1), exp (1), per (1), bur (1), bot (1), std (2), exp (2), per (2), bur (2), bot (2),
xtick style = draw=none,
ytick = 0, 0.5,1,
boxplot/every median/.style=red,ultra thick
in 1,...,10
+[boxplot, fill, draw=black, mark = *] table[y index=] ;

GPT-2 detector

[col sep=comma]data/openai_scores.csv
boxplot/draw direction = y,
xtick = 1, 2, 3, 4,5,6,7,8,9,10,
xticklabel style = align=center, font=, rotate=30,
xticklabels = std (1), exp (1), per (1), bur (1), bot (1), std (2), exp (2), per (2), bur (2), bot (2),
xtick style = draw=none,
ytick = 0, 0.5,1,
boxplot/every median/.style=red,ultra thick
in 1,...,10
+[boxplot, fill, draw=black, mark = *] table[y index=] ;

[col sep=comma]data/turnitin_scores.csv
boxplot/draw direction = y,
xtick = 1, 2, 3, 4,5,6,7,8,9,10,
xticklabel style = align=center, font=, rotate=30,
xticklabels = std (1), exp (1), per (1), bur (1), bot (1), std (2), exp (2), per (2), bur (2), bot (2),
xtick style = draw=none,
ytick = 0, 0.5,1,
boxplot/every median/.style=red,ultra thick
in 1,...,10
+[boxplot, fill, draw=black, mark = *] table[y index=] ;

Influence of perplexity and burstiness prompts on detection (number of prompts in brackets; std = standard prompt; exp = prompt to explain and increase burstiness & perplexity; per = increase perplexity; bur = increase burstiness; bot = increase both)

§.§ Character-level mutations

The influence of the character-level mutations on the detection scores is shown in Table <ref>. For the GPT-2 detector, all three replacements lead to a detection score of 0. For Open AI, all attacks lowered the detection score, although not as much as for the GPT-2 detector. Turnitin detected the replacement of Latin characters with Cyrillic characters and flagged the attack. The substitution of ls with capital is, however, remained undetected and significantly lowered the detection score, therefore presenting a successful attack strategy.

GPT-2      OpenAI     Turnitin
Standard                              0.67       0.77         0.75

Swap a lat.-cyr.               0          0.52         x

Swap e lat.-cyr.              0          0.48         x

Swap l - I         0          0.38          0.21

Character-level mutation mean detection score

§ CONCLUSION

This study explored the effectiveness of resource-efficient adversarial attacks on neural text detectors, based on texts generated by GPT-3.5 and the three detectors GPT-2 output detector, OpenAI classifier, and Turnitin. Of the three investigated strategies, parameter tweaking and character-level mutations were successful for all three detectors. Prompt engineering was only successful for the GPT-2 output detector. All strategies are resource efficient and easy to implement, effectively showing that currently available detectors cannot reliably detect AI-generated texts and are vulnerable to adversarial attacks.

§ APPENDIX - PROMPTS

§.§ Advanced Prompt

“Write a five-hundred-word argumentative essay on the topic ‘topic’. Include personal reflections, use a mix of long and short sentences, employ rhetorical questions to engage the reader, maintain a conversational tone in parts, and play around with the paragraph structure to create a dynamic and engaging piece of writing. Try to include factual and contextual information, use advanced concepts and vocabulary. Utilize a combination of complex and simple vocabulary. Try to mimic human writing as closely as you can. Avoid passive voice, as it tends to occur more often in AI-generated texts. Add a few examples from the real world illustrating your point.”

§.§ Perplexity and Burstiness

“Write a five-hundred-word argumentative essay on the topic ‘topic’. When it comes to writing content, two factors are crucial, perplexity and burstiness. Perplexity measures the complexity of the text. Separately, burstiness compares the variations of sentences. Humans tend to write with greater burstiness, for example, with some longer or more complex sentences alongside shorter ones. AI sentences tend to be more uniform. Therefore, when writing the following content, I need it to have a good amount of perplexity and burstiness.


# Inverse problem of determining an order of the Riemann-Liouville time-
fractional derivative

Shavkat Alimov  and  Ravshan Ashurov  National University of Uzbekistan named
after Mirzo Ulugbek and Institute of Mathematics, Uzbekistan Academy of
Science Institute of Mathematics, Uzbekistan Academy of Science, Tashkent, 81
Mirzo Ulugbek str. 100170<EMAIL_ADDRESS>
# Inverse problem of determining an order of the Riemann-Liouville time-
fractional derivative

Shavkat Alimov  and  Ravshan Ashurov  National University of Uzbekistan named
after Mirzo Ulugbek and Institute of Mathematics, Uzbekistan Academy of
Science Institute of Mathematics, Uzbekistan Academy of Science, Tashkent, 81
Mirzo Ulugbek str. 100170<EMAIL_ADDRESS>
###### Abstract.

The inverse problem of determining the order of the fractional Riemann-
Liouville derivative with respect to time in the subdiffusion equation with an
arbitrary positive self-adjoint operator having a discrete spectrum is
considered. Using the classical Fourier method it is proved, that the value of
the norm $||u(t)||$ of the solution at a fixed time instance recovers uniquely
the order of derivative. A list of examples is discussed, including a linear
system of fractional differential equations, differential models with
involution, fractional Sturm-Liouville operators, and many others.

AMS 2000 Mathematics Subject Classifications : Primary 35R11; Secondary 74S25.
Key words: Determination of order of derivative, Fourier method, inverse and
initial-boundary value problem, Riemann-Liouville derivatives, subdiffusion
equation.

##  1\. Main result

It has long been known that to model subdiffusion (anomalous or slow
diffusion) processes, it is necessary to use differential equations of
fractional order $\rho\in(0,1)$. But in this case, unlike differential
equations of integer order, an order of the fractional derivative $\rho$ is
often unknown and difficult to be directly measured. The determination of this
parameter is called the inverse problem of determining the order of the
fractional derivative. These inverse problems are not only theoretically
interesting, but also necessary for finding solutions to initial-boundary
value problems and studying properties of solutions. The paper [1] by Li, Liu,
Yamamoto surveys works on such inverse problems.

In the present paper, we are concerned with inversion for order in the
subdiffusion equation with the Riemann-Liouville time-fractional derivative.

Let $H$ be a separable Hilbert space with the scalar product $(\cdot,\cdot)$
and the norm $||\cdot||$ and $A:H\rightarrow H$ be an arbitrary positive
selfadjoint operator in $H$. Suppose that $A$ has a complete in $H$ system of
orthonormal eigenfunctions $\\{\upsilon_{k}\\}$ and a countable set of
nonnegative eigenvalues $\lambda_{k}$. It is convenient to assume that the
eigenvalues do not decrease as their number increases, i.e.
$0<\lambda_{1}\leq\lambda_{2}\cdot\cdot\cdot$.

Using the definitions of a strong integral and a strong derivative, fractional
analogues of integrals and derivatives can be determined for vector-valued
functions (or simply functions) $h:\mathbb{R}_{+}\rightarrow H$, while the
well-known formulae and properties are preserved (see, for example,
[2]).Recall that the fractional integration of order $\rho<0$ of the function
$h(t)$ defined on $[0,\infty)$ has the form

$\partial_{t}^{\rho}h(t)=\frac{1}{\Gamma(-\rho)}\int\limits_{0}^{t}\frac{h(\xi)}{(t-\xi)^{\rho+1}}d\xi,\quad
t>0,$

provided the right-hand side exists. Here $\Gamma(\rho)$ is Euler’s gamma
function. Using this definition one can define the Riemann - Liouville
fractional derivative of order $\rho$, $0<\rho<1$, as

$\partial_{t}^{\rho}h(t)=\frac{d}{dt}\partial_{t}^{\rho-1}h(t).$

If in this definition we interchange the differentiation and fractional
integration, then we get the definition of a regularized derivative, that is,
the definition of a fractional derivative in the sense of Caputo:

$D_{t}^{\rho}h(t)=\partial_{t}^{\rho-1}\frac{d}{dt}h(t).$

Note that if $\rho=1$ , then fractional derivative coincides with the ordinary
classical derivative of the first order:

$\partial_{t}h(t)=\frac{d}{dt}h(t).$

Let $\rho\in(0,1)$ be a fixed number and let $C((a,b);H)$ stand for a set of
continuous functions $u(t)$ of $t\in(a,b)$ with values in $H$. Consider the
Cauchy type problem:

(1.1) $\left\\{\begin{aligned} &\partial_{t}^{\rho}u(t)+Au(t)=0,\quad 0<t\leq
T;\\\ &\lim\limits_{t\rightarrow
0}\partial_{t}^{\rho-1}u(t)=\varphi,\end{aligned}\right.$

where $\varphi$ is a given vector in $H$. If $\rho=1$, then the initial
condition has the form $u(0)=\varphi$. This problem is called a _forward
problem_.

######  Definition 1.1.

A function $u(t)$ with the properties $\partial_{t}^{\rho}u(t),Au(t)\in
C((0,T];H)$ and satisfying conditions (1.1) is called the solution of the
forward problem (1.1).

Let us denote by $E_{\rho,\mu}(t)$ the Mittag-Leffler function of the form

$E_{\rho,\mu}(t)=\sum\limits_{k=0}^{\infty}\frac{t^{k}}{\Gamma(\rho k+\mu)}.$

We first prove the existence and uniqueness of a solution of problem (1.1).

######  Theorem 1.2.

For any $\varphi\in H$ problem (1.1) has a unique solution and this solution
has the form

(1.2)
$u(t)=\sum\limits_{k=1}^{\infty}t^{\rho-1}E_{\rho,\rho}(-\lambda_{k}t^{\rho})(\varphi,v_{k})v_{k}.$

The problem (1.1) for various operators A has been considered by a number of
authors. Let us mention only some of these works. The case of one spatial
variable $x\in\mathbb{R}$ and subdiffusion equation with $Au=u_{xx}$
considered, for example, in the book of A.A. Kilbas et al. [3] and monograph
of A. V. Pskhu [4], and references in these works. The paper Gorenflo, Luchko
and Yamamoto [5] is devoted to the study of subdiffusion equations in Sobelev
spaces. In the paper by Kubica and Yamamoto [6], initial-boundary value
problems for equations with time-dependent coefficients are considered. In the
multidimensional case $(x\in R^{N})$, instead of the differential expression
$u_{xx}$, authors of the papers [3], [7]-[9] considered the Laplace operator
and Umarov [10] consedered pseudodifferential operators with constant
coefficients in the whole space $\mathbb{R}^{N}$.

A result similar to the above, in the case when the fractional part of the
equation (1) is the Caputo derivative, was obtained by M. Ruzhansky et al.
[11]. In the case when $A$ is an arbitrary elliptic differential operator,
this theorem was proved in [12].

Obviously solution (1.2) depends on $\rho\in(0,1)$. Now let us consider the
order of fractional derivative $\rho$ as a unknown parameter and consider an
inverse problem: can we identify uniquely this parameter $\rho$, if we have as
a additional information the norm

(1.3) $W(t_{0},\rho)=||u(t_{0})||^{2}=d_{0}$

at a fixed time instant $t_{0}>0$?

Problem (1.1) together with extra condition (1.3) is called _the inverse
problem_.

To solve this inverse problem we fix a number $\rho_{0}\in(0,1)$ and consider
the problem for $\rho\in[\rho_{0},1)$.

######  Definition 1.3.

A pair $\\{u(t),\rho\\}$ of the solution $u(t)$ to the forward problem and the
parameter $\rho\in[\rho_{0},1]$ satisfying the additional condition (1.3) is
called the solution of the inverse problem.

######  Lemma 1.4.

Given $\rho_{0}$ from interval $0<\rho_{0}<1$, there exists a number
$T_{0}=T_{0}(\rho_{0},\lambda_{1})$, such that for all $t_{0}\geq T_{0}$ and
for arbitrary $\varphi\in H$ function $W(t_{0},\rho)$ decreases monotonically
with respect to $\rho\in[\rho_{0},1]$.

The main result of the paper is the following:

######  Theorem 1.5.

Let $\varphi\in H$ and $t_{0}\geq T_{0}$. Then the inverse problem has a
unique solution $\\{u(t),\rho\\}$ if and only if

$W(t_{0},1)\leq d_{0}\leq W(t_{0},\rho_{0}).$

Theorem 1.5 gives a positive answer to the problem posed in review article by
Z. Li et al. [1] (p. 440) in the Conclusions and Open Problems section: Is it
possible to identify uniquely the order of fractional derivatives if an
additional information about the solution is specified at a fixed time instant
as ”the observation data”?.

Note that our result shows, that it is possible to restore the order of the
fractional derivative by using the value of $W(t,\rho)$ at a fixed time
instant $t_{0}$ as ”the observation data”.

The inverse problem of determining the order of time fractional derivative in
subdiffusion equations has been studied by a number of authors (see a survey
paper [1] and references therein, [13]-[23]). It is necessary to note that in
all these publications the following relation was taken as an additional
condition

(1.4) $u(x_{0},t)=h(t),\,\,0<t<T,$

at a monitoring point $x_{0}\in\overline{\Omega}$. But this condition, as a
rule, can guarantee only the uniqueness of the solution of the inverse problem
(see [13]-[16]). However, as Theorem 2 states, unlike (1.4), condition (1.3)
guarantees both uniqueness and the existence of a solution.

Hatano et al. [17] considered the equation $\partial_{t}^{\rho}u=\triangle u$
with the Dirichlet boundary condition and the initial function $\varphi(x)$
(see also [18]). They proved the following property of the parameter $\rho$:
if $\varphi\in C_{0}^{\infty}(\Omega)$ and $\triangle\varphi(x_{0})\neq 0$,
then

$\rho=\lim\limits_{t\rightarrow
0}\big{[}t\partial_{t}u(x_{0},t)[u(x_{0},t)-\varphi(x_{0})]^{-1}\big{]}.$

For the best of our knowledge, only in the paper [19] by J. Janno the
existence problem is considered. Giving an extra boundary condition
$Bu(\cdot,t)=h(t),0<t<T$ the author succeeded to prove the existence theorem
for determining the order $\rho$, $0<\rho<1$, of the Caputo derivative and the
kernel of the integral operator in the equation.

We also note the following recent papers. In the paper by Z. Li and Z. Zhang
[20] the authors studied the uniqueness in an inverse problem for
simultaneously determining the order of time fractional derivative and a
source function in a subdiffusion equation. In [21], M. Yamamoto proved the
uniqueness in determining both orders of fractional time derivatives and
spatial derivatives in diffusion equations. The proof relies on the
eigenfunction expansion and the asymptotics of the Mittag-Leffler function.
The authors of [22] discuss similar issues discussed in the present paper. As
an additional information for inverse problem they have considered the value
of projection of the solution onto the first eigenfunction at a fixed time
instance. Note, that results of paper [22] are applicable only in case, when
the first eigenvalue of the corresponding elliptic operator is equal to zero.
We also mention the paper [23], in which a result similar to Theorem 2 was
proved for the subdiffusion equation with the Caputo derivative.

In conclusion, we give the following remarks:

1) As the operator $A$, one can take any equations of mathematical physics
considered in Section 6 of the article by M. Ruzhansky et al. [11], including
the classical Sturm-Liouville problem, differential models with involution,
fractional Sturm-Liouville operators, harmonic and anharmonic oscillators,
Landau Hamiltonians, fractional Laplacians, harmonic and anharmonic operators
on the Heisenberg group.

It should be noted, that the authors of [11] considered inverse problems for
restoring the right-hand side of a subdiffusion equation for a large class of
positive operators $A$.

2) Further, let us take $\mathbb{R}^{N}$ as a Hilbert space $H$ and
$N$-dimensional symmetric quadratic matrix $A=\\{a_{i,j}\\}$ with constant
elements $a_{i,j}$ as operator $A$. In this case, the problem (1.1) coincides
with the Cauchy problem for a linear system of fractional differential
equations.

3) You can also consider various options for the function $W(t,\rho)$.
Examples $W(t,\rho)=||Au(t)||^{2}$, $W(t,\rho)=(u,\varphi)$.

##  2\. Forward problem

In the present section we prove Theorem 1.2.

To prove the existence of the forward problem’s solution we remind the
following estimate of the Mittag-Leffler function with a negative argument
(see, for example, [24], p. 136)

(2.1) $|E_{\rho,\mu}(-t)|\leq\frac{C}{1+t},\quad t>0.$

In accordance with Definition 1.1, we will first show that for function (1.2)
one has $Au(t)\in C((0,T];H)$. To do this, consider the sum

$S_{j}(t)=\sum\limits_{k=1}^{j}t^{\rho-1}E_{\rho,\rho}(-\lambda_{k}t^{\rho})\,(\varphi,v_{k})v_{k}.$

Then

$AS_{j}(t)=\sum\limits_{k=1}^{j}\lambda_{k}t^{\rho-1}E_{\rho}(-\lambda_{k}t^{\rho})\,(\varphi,v_{k})v_{k}.$

Due to the Parseval equality we may write

$||AS_{j}(t)||^{2}=\sum\limits_{k=1}^{j}|\lambda_{k}t^{\rho-1}E_{\rho}(-\lambda_{k}t^{\rho})\,(\varphi,v_{k})|^{2}\leq
Ct^{-2}||\varphi||^{2}.$

Here we used estimate (2.1) and the inequality $\lambda t^{\rho}(1+\lambda
t^{\rho})^{-1}<1$.

Hence, we obtain $Au(t)\in C((0,T];H)$.

Further, from equation (1.1) one has $\partial_{t}^{\rho}S_{j}(t)=-AS_{j}(t)$.
Therefore, from above reasoning, we finally have $\partial_{t}^{\rho}u(t)\in
C((0,T];H)$.

It is not hard to verify the fulfillment of equation (1.1) (see, for example,
[25], p. 173 and [26]) and the initial condition therein.

Now we prove the uniqueness of the forward problem’s solution.

Suppose that problem (1.1) has two solutions $u_{1}(t)$ and $u_{2}(t)$. Our
aim is to prove that $u(t)=u_{1}(t)-u_{2}(t)\equiv 0$. Since the problem is
linear, then we have the following homogenous problem for $u(t)$:

(2.2) $\partial_{t}^{\rho}u(t)+Au(t)=0,\quad t>0;$ (2.3)
$\lim\limits_{t\rightarrow 0}\partial_{t}^{\rho-1}u(t)=0.$

Set

$w_{k}(t)\ =\ (u(t),v_{k}).$

It follows from (2.2) that for any $k\in\mathbb{N}$

$\partial_{t}^{\rho}w_{k}(t)=(\partial_{t}^{\rho}u(t),v_{k})=-\,(Au(t),v_{k})=-\,(u(t),Av_{k})=-\,\lambda_{k}w_{k}(t).$

Therefore, we have the following Cauchy problem for $w_{k}(t)$ (see (2.3)):

$\partial_{t}^{\rho}w_{k}(t)+\lambda_{k}w_{k}(t)=0,\quad
t>0;\quad\lim\limits_{t\rightarrow 0}\partial_{t}^{\rho-1}w_{k}(t)=0.$

This problem has the unique solution (see, for example, [25], p. 173 and
[26]). Therefore, $w_{k}(t)=0$ for $t>0$ and for all $k\geq 1$. Then by the
Parseval equation we obtain $u(t)=0$ for all $t>0$. Hence uniqueness of the
solution is proved.

Thus the proof of Theorem 1.2 is complete.

##  3\. Inverse problem

######  Lemma 3.1.

Given $\rho_{0}$ from the interval $0<\rho_{0}<1$, there exists a number
$T_{0}=T_{0}(\rho_{0},\lambda_{1})$, such that for all $t_{0}\geq T_{0}$ and
$\lambda\geq\lambda_{1}$ functions
$e_{\lambda}(\rho)=t_{0}^{\rho-1}E_{\rho,\rho}(-\lambda t_{0}^{\rho})$ are
positive and they decrease monotonically with respect to
$\rho\in[\rho_{0},1]$.

###### Proof.

Let us denote by $\delta(1;\beta)$ a contour oriented by non-decreasing
$\arg\zeta$ consisting of the following parts: the ray $\arg\zeta=-\beta$,
$|\zeta|\geq 1$, the arc $-\beta\leq\arg\zeta\leq\beta$, $|\zeta|=1$, and the
ray $\arg\zeta=\beta$, $|\zeta|\geq 1$. If $0<\beta<\pi$, then the contour
$\delta(1;\beta)$ divides the complex $\zeta$-plane into two unbounded parts,
namely $G^{(-)}(1;\beta)$ to the left of $\delta(1;\beta)$ by orientation, and
$G^{(+)}(1;\beta)$ to the right of it. The contour $\delta(1;\beta)$ is called
the Hankel path.

Let $\beta=\frac{3\pi}{4}\rho$, $\rho\in[\rho_{0},1)$. Then by the definition
of this contour $\delta(1;\beta)$, we arrive at (see [17], formula (2.29), p.
135, note $-\lambda t_{0}^{\rho}\in G^{(-)}(1;\beta)$)

(3.1) $t_{0}^{\rho-1}E_{\rho,\rho}(-\lambda
t_{0}^{\rho})=-\frac{1}{\lambda^{2}t_{0}^{\rho+1}\Gamma(-\rho)}+\frac{\rho}{2\pi
i\lambda^{2}t_{0}^{\rho+1}}\int\limits_{\delta(1;\beta)}\frac{e^{\zeta^{1/\rho}}\zeta^{\frac{1}{\rho}+1}}{\zeta+\lambda
t_{0}^{\rho}}d\zeta=f_{1}(\rho)+f_{2}(\rho).$

To prove the lemma, it is suffices to show that the derivative
$\frac{d}{d\rho}e_{\lambda}(\rho)$ is negative for all $\rho\in[\rho_{0},1)$,
since the positivity of $e_{\lambda}(\rho)$ follows from the inequality
$e_{\lambda}(1)=e^{-\lambda t}>0$.

It is not hard to estimate the derivative $f^{\prime}_{1}(\rho)$. Indeed, let
$\Psi(\rho)$ be the logarithmic derivative of the gamma function
$\Gamma(\rho)$ (for the definition and properties of $\Psi$ see [20]). Then
$\Gamma^{\prime}(\rho)=\Gamma(\rho)\Psi(\rho)$, and therefore,

$f_{1}^{\prime}(\rho)=\frac{\ln
t_{0}-\Psi(-\rho)}{\lambda^{2}t_{0}^{\rho+1}\Gamma(-\rho)}.$

Since

$\frac{1}{\Gamma(-\rho)}=-\frac{\rho}{\Gamma(1-\rho)}=-\frac{\rho(1-\rho)}{\Gamma(2-\rho)},\quad\Psi(-\rho)=\Psi(1-\rho)+\frac{1}{\rho}=\Psi(2-\rho)+\frac{1}{\rho}-\frac{1}{1-\rho},$

the function $f_{1}^{\prime}(\rho)$ can be represented as follows

(3.2)
$f_{1}^{\prime}(\rho)=\frac{1}{\lambda^{2}t_{0}^{\rho+1}}\frac{\rho(1-\rho)[\Psi(2-\rho)-\ln
t_{0}]+1-2\rho}{\Gamma(2-\rho)}=-\frac{f_{11}(\rho)}{\lambda^{2}t_{0}^{\rho+1}\Gamma(2-\rho)}.$

If $\gamma\approx 0,57722$ is the Euler-Mascheroni constant, then
$\Psi(2-\rho)<1-\gamma$. Therefore,

$f_{11}(\rho)>\rho(1-\rho)[\ln t_{0}-(1-\gamma)])+2\rho-1.$

For $t_{0}=e^{1-\gamma}e^{2/\rho}$ one has $\rho(1-\rho)[\ln
t_{0}-(1-\gamma)])+2\rho-1=1$. Hence, $f_{11}(\rho)\geq 1$, provided
$t_{0}\geq T_{0}$ and

(3.3) $T_{0}=e^{1-\gamma}e^{2/\rho_{0}}.$

Thus, by virtue of (3.2), for all such $t_{0}$ we arrive at

(3.4) $f_{1}^{\prime}(\rho)\leq-\frac{1}{\lambda^{2}t_{0}^{\rho+1}}.$

To estimate the derivative $f^{\prime}_{2}(\rho)$, we denote the integrand in
(3.1) by $F(\zeta,\rho)$:

$F(\zeta,\rho)=\frac{1}{2\pi
i\rho\lambda^{2}t_{0}^{\rho+1}}\cdot\frac{e^{\zeta^{1/\rho}}\zeta^{1/\rho+1}}{\zeta+\lambda
t_{0}^{\rho}}.$

Note, that the domain of integration $\delta(1;\beta)$ also depends on $\rho$.
To take this circumstance into account when differentiating the function
$f^{\prime}_{2}(\rho)$, we rewrite the integral (3.1) in the form:

$f_{2}(\rho)=f_{2+}(\rho)+f_{2-}(\rho)+f_{21}(\rho),$

where

$f_{2\pm}(\rho)=e^{\pm i\beta}\int\limits_{1}^{\infty}F(s\,e^{\pm
i\beta},\rho)\,ds,$

$f_{21}(\rho)=i\int\limits_{-\beta}^{\beta}F(e^{iy},\rho)\,e^{iy}dy=i\beta\int\limits_{-1}^{1}F(e^{i\beta
s},\rho)\,e^{i\beta s}ds.$

Let us consider the function $f_{2+}(\rho)$. Since $\beta=\frac{3\pi}{4}\rho$
and $\zeta=s\,e^{i\beta}$, then

$e^{\zeta^{1/\rho}}=e^{\frac{1}{\sqrt{2}}(i-1)s^{\frac{1}{\rho}}},$

The derivative of the function $f_{2+}(\rho)$ has the form

$f_{2+}^{\prime}(\rho)={I}\cdot\int\limits_{1}^{\infty}\frac{e^{\frac{1}{\sqrt{2}}(i-1)s^{\frac{1}{\rho}}}s^{1/\rho+1}e^{2ia\rho}\big{[}\frac{1}{\rho^{2}}(\frac{1}{\sqrt{2}}(1-i)s^{\frac{1}{\rho}}-1)\ln
s+2ia-\frac{1}{\rho}-\ln t_{0}-\frac{iase^{ia\rho}+\lambda t_{0}^{\rho}\ln
t_{0}}{se^{ia\rho}+\lambda t_{0}^{\rho}}\big{]}}{se^{ia\rho}+\lambda
t_{0}^{\rho}}ds.$

where $I=e^{ia}(2\pi i\rho\lambda^{2}t_{0}^{\rho+1})$ and $a=\frac{3\pi}{4}$.
By virtue of the inequality $|se^{ia\rho}+\lambda t_{0}^{\rho}|\geq\lambda
t_{0}^{\rho}$ we arrive at

$|f_{2+}^{\prime}(\rho)|\leq\frac{C}{\rho\lambda^{3}t_{0}^{2\rho+1}}\cdot\int\limits_{1}^{\infty}e^{-\frac{1}{2}s^{\frac{1}{\rho}}}s^{1/\rho+1}\big{[}\frac{1}{\rho^{2}}s^{\frac{1}{\rho}}\ln
s+\ln t_{0}\big{]}ds.$

######  Lemma 3.2.

Let $0<\rho\leq 1$ and $m\in\mathbb{N}$. Then

$I(\rho)=\frac{1}{\rho}\int\limits_{1}^{\infty}e^{-\frac{1}{2}s^{\frac{1}{\rho}}}s^{\frac{m}{\rho}+1}ds\leq
C_{m}.$

###### Proof.

Set $r=s^{\frac{1}{\rho}}$. Then

$s=r^{\rho},\quad ds=\rho r^{\rho-1}dr.$

Therefore,

$I(\rho)=\int\limits_{1}^{\infty}e^{-\frac{1}{2}r}r^{m-1+2\rho}dr\leq\int\limits_{1}^{\infty}e^{-\frac{1}{2}r}r^{m+1}dr=C_{m}.$

∎

Lemma 3.2 is proved.

Application of this lemma gives (note, $\frac{1}{\rho}lns<s^{\frac{1}{\rho}}$,
provided $s\geq 1$)

$|f_{2+}^{\prime}(\rho)|\leq\frac{C}{\lambda^{3}t_{0}^{2\rho+1}}\big{[}\frac{C_{3}}{\rho}+C_{1}lnt_{0}\big{]}\leq\frac{C}{\lambda^{3}t_{0}^{2\rho+1}}\big{[}\frac{1}{\rho}+lnt_{0}\big{]}.$

Function $f_{2-}(\rho)$ has exactly the same estimate.

Now consider the function $f_{21}(\rho)$. For its derivative we have

$f_{21}^{\prime}(\rho)=\frac{a}{2\pi
i\lambda^{2}t_{0}^{\rho+1}}\cdot\int\limits_{-1}^{1}\frac{e^{e^{ias}}e^{ias}e^{2ia\rho
s}\big{[}2ias-\ln t_{0}-\frac{iase^{ia\rho}+\lambda t_{0}^{\rho}\ln
t_{0}}{e^{ia\rho s}+\lambda t_{0}^{\rho}}\big{]}}{e^{ia\rho s}+\lambda
t_{0}^{\rho}}ds.$

Therefore,

$|f_{21}^{\prime}(\rho)|\leq C\frac{lnt_{0}}{\lambda^{3}t_{0}^{2\rho+1}}.$

Taking into account stimmte (3.4) and the estimates of $f_{2\pm}^{\prime}$ and
$f_{21}^{\prime}$, we have

$\frac{d}{d\rho}e_{\lambda}(\rho)<-\frac{1}{\lambda^{2}t_{0}^{\rho+1}}+C\frac{1/\rho+\ln
t_{0}}{\lambda^{3}t_{0}^{2\rho+1}}.$

In other words, this derivative is negative if

$t_{0}^{\rho_{0}}>\frac{C}{\lambda_{0}}(\frac{1}{\rho_{0}}+\ln t_{0}).$

Hence, there exists a number $T_{0}=T_{0}(\lambda_{0},\rho_{0})$ (see also
(3.3)) such, that for all $t_{0}\geq T_{0}$

$\frac{d}{d\rho}[{t_{0}^{\rho-1}E_{\rho,\rho}(-\lambda
t_{0}^{\rho})}]<0,\quad\lambda\geq\lambda_{0},\quad\rho\in[\rho_{0},1].$

Lemma 3.1 is proved. ∎

Since

$W(t,\rho)=||u(t)||^{2}=\sum\limits_{k=1}^{\infty}|(\varphi,v_{k})|^{2}|t^{\rho-1}E_{\rho}(-\lambda_{k}t^{\rho})|^{2},$

then Lemma 1.4 follows immediately from Lemma 3.1. Theorem 1.5 is an easy
consequence of these two lemmas.

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[a]Yigal Shamir

#  $S$ parameter from a prototype composite-Higgs model

Maarten Golterman  William I. Jay  Ethan T. Neil  Benjamin Svetitsky

###### Abstract

We have calculated the low-energy constant $L_{10}$ in a prototype composite
Higgs model with dynamical fermions in two different representations of the
gauge group. The resulting contribution of the new strong sector to the $S$
parameter is consistent with current bounds on the vacuum misalignment
parameter. We end with a brief discussion of future directions.

##  1 Introduction

According to the composite Higgs paradigm, the Higgs boson is a pseudo Nambu-
Goldstone boson of a new strong interaction [2, 3]. This protects its mass
from receiving large radiative corrections. Hypercolor—the new strong
interaction—may be operative at the few TeV scale (see Refs. [4, 5, 6] for
reviews). One can also use the hypercolor interaction to generate a partially-
composite top quark via linear coupling to a hypercolor baryon [7].

We have been studying a prototype composite-Higgs model using lattice
techniques [8, 9, 10, 11, 12, 13]. The model is an SU(4) gauge theory with 2
Dirac fermions in the fundamental representation, together with 2 Dirac
fermions in the sextet, or two-index antisymmetric, representation. Since the
sextet is real, this is equivalent to 4 Majorana fermions. Our work was the
first lattice calculation with dynamical fermions in two different
representations. For similar lattice calculations, see Refs. [14, 15]. For
lattice work related to composite-Higgs models based on an Sp(4) gauge theory,
see Refs. [16, 17, 18, 19].

In itself, the fermion content of the prototype model is not enough to meet
phenomenological requirements. But it is quite close to one of the composite-
Higgs models proposed by Ferretti and Karateev [20], labeled M6 in Ref. [21],
which contains 3 fundamental Dirac fermions and 5 sextet Majorana fermions
(see also Refs. [22, 23]). Note that the Ferretti-Karateev models are
partially, but not fully, ultraviolet complete: the origin of the 4-fermion
couplings needed to generate a partially composite top quark remains
unspecified. For a proposal for a full ultraviolet completion, see Ref. [24].

Here we report on our calculation of the low-energy constant $L_{10}$ [13],
which largely follows the QCD calculations of Refs. [25, 26]. We have
calculated $L_{10}$ in the sextet sector of our two-representation theory,
since in the M6 model the electro-weak symmetries are embedded in the unbroken
global symmetry of that sector. Our result for $L_{10}$ allows us to constrain
the contribution of the hypercolor theory to the $S$ parameter in terms of the
vacuum misalignment parameter [4, 5, 6].

##  2 Chiral perturbation theory

In the continuum, the correlator of a left-handed and a right-handed current
is decomposed as

$\left\langle
J_{L\mu}J_{R\nu}\right\rangle=(q^{2}\delta_{\mu\nu}-q_{\mu}q_{\nu})\Pi^{(1)}(q^{2})+q_{\mu}q_{\nu}\Pi^{(0)}(q^{2})\
,$ (1)

with transverse function $\Pi^{(1)}$ and longitudinal function $\Pi^{(0)}$.
The transverse function is given by

$\Pi^{(1)}=\frac{F^{2}}{q^{2}}+\hat{\Pi}(q^{2})\ ,$ (2)

where the leading-order pole reflects a kinematical singularity. To next-to-
leading order (NLO) in chiral perturbation theory and for $N$ Majorana
fermions [27, 28]

$\hat{\Pi}(q^{2})=\frac{N+2}{96\pi^{2}}\left[\frac{1}{3}+\log\left(\frac{M^{2}}{\mu^{2}}\right)-H(s)\right]+8L_{10}\
,$ (3)

which includes the dependence on $L_{10}$. Here $M$ is the pion mass and $\mu$
is the renormalization scale. The momentum dependence enters via

$H(s)=2s^{2}+s^{3}\log\left(\frac{s-1}{s+1}\right)\ ,$ (4)

with $s=\sqrt{1+4M^{2}/q^{2}}$. The function $H(s)$ has no free parameters. We
will also need the difference

$\Pi^{(1-0)}=\Pi^{(1)}-\Pi^{(0)}\ =\
\frac{F^{2}}{q^{2}+M^{2}}+\hat{\Pi}(q^{2})\ .$ (5)

Turning to the lattice calculation, our ensembles were generated with
dynamical Wilson-clover fermions for both representations [8]. nHYP smearing
was used in the fermion action [29, 30], and an NDS term was added to the
gauge action to further improve the performance of the smeared links [31]. In
view of the importance of chiral symmetry for the calculation of $L_{10}$ we
used staggered valence fermions, i.e., we did a mixed-action calculation.
$J_{L\mu}$ and $J_{R\nu}$ were constructed from the standard vector and axial
staggered currents, except again with nHYP links. We made the usual
replacement $q_{\mu}\to\hat{q}_{\mu}=(2/a)\sin(aq_{\mu}/2)$, with $a$ the
lattice spacing. The renormalization scale was $1/\mu^{2}=t_{0}$, where
$t_{0}$ is the gradient-flow scale.

The pole terms, i.e., the first term on the right-hand sides of Eqs. (2) and
(5), involve the pure valence pion, so that $M=M_{vv}$ and $F=F_{vv}$. At NLO,
the pion in the loop is a mixed sea-valence pion, hence, in Eq. (3), as well
as inside the argument $s$ of $H(s)$, $M=M_{vs}$, with

$M_{vs}^{2}=\frac{M_{ss}^{2}+M_{vv}^{2}}{2}+\frac{a^{2}}{t_{0}^{2}}\,\Delta_{\rm
mix}\ .$ (6)

Here $\Delta_{\rm mix}\geq 0$ is a new parameter peculiar to the mixed-action
setting [32, 33, 34, 35]. In order to explore the possible effects of the next
order in chiral perturbation theory, we have augmented Eq. (3) by the (NNLO)
analytic terms,

$t_{0}\left(b_{q}\,q^{2}+b_{ss}\,M_{ss}^{2}+b_{vv}\,M_{vv}^{2}\right)+b_{a}\,\frac{a^{2}}{t_{0}}\
.$ (7)

##  3 Results

We used twelve $16^{3}\times 32$ ensembles and three $24^{3}\times 48$
ensembles taken from Ref. [8], with gradient-flow scale $t_{0}/a^{2}$ in the
range 0.9–2.7, and sea-pion mass $\sqrt{t_{0}}M_{ss}$ in the range 0.2–0.58.
After some exploratory studies, we settled on a calculation with 7 valence
masses $am_{v}$ in the range 0.01–0.05. We successfully fitted $\Pi^{(1-0)}$,
whose correlations turned out to be smaller than those of $\Pi^{(1)}$, always
including $L_{10}$ and $b_{vv}$ in the fit. In order to limit other sources of
higher-order corrections apart from the valence mass, we evaluated the
correlators only at the smallest time-like momentum. We tried all 16
combinations of the 4 remaining parameters ($\Delta_{\rm mix}$, $b_{q}$,
$b_{a}$ and $b_{ss}$), always obtaining a good $p$-value.

Figure 1:  Sixteen fits of $\Pi^{(1-0)}$ to data from all 7 valence masses.
All fits include $L_{10}$ and $b_{vv}$ as parameters but have different
combinations of $\Delta_{\rm mix}$ and the other NNLO parameters $b_{q}$,
$b_{ss}$ and $b_{a}$. Fits without $\Delta_{\rm mix}$ are shown in purple, and
with $\Delta_{\rm mix}$ in orange.

The results are shown in Fig. 1. The NLO mixed-action parameter $\Delta_{\rm
mix}$ was always consistent with zero, so we focus on the fits without
$\Delta_{\rm mix}$. The value of $b_{vv}$ was stable and more than $5\sigma$
away from zero, which explains why this NNLO parameter had to be included in
all the fits. Fit No. 1 includes only $L_{10}$ and $b_{vv}$, and has a very
small statistical error (about the size of the data point). The remaining 7
fits with $\Delta_{\rm mix}=0$ include different subsets of the other NNLO
parameters $b_{q}$, $b_{a}$ and $b_{ss}$. They give rise to considerable
variation in the value of $L_{10}$, as well as to increasing statistical
error. The results of fits 3, 5, and 9, which include only a single additional
NNLO parameter suggest that our main source of uncertainty is systematic. We
used the central values of these fits to bracket the systematic error of our
final result, and their statistical errors to estimate the statistical error
of this result, obtaining

$L_{10}=-0.0100(12)_{\rm stat}(35)_{\rm syst}\ .$ (8)

The central value of our final result coincides with that of fit 1, which
contains only $L_{10}$ and $b_{vv}$. Changing the renormalization scale $\mu$
in Eq. (3) from $1/\sqrt{t_{0}}$ to the sextet vector meson mass shifts the
central value of $L_{10}$ by about $-0.00035$, a 3.5% shift.

The contribution of the hypercolor sector to the $S$ parameter is

$S_{\rm HC}=\xi S_{\rm NLO}\ .$ (9)

Here $\xi=2v^{2}/F_{6}^{2}$ is the vacuum misalignment parameter, where
$v=246$ GeV is the vacuum expectation value of the Higgs field in the Standard
Model, and $F_{6}$ is the decay constant of Nambu-Goldstone bosons made of the
sextet fermions in the chiral limit. $S_{\rm NLO}$ is given by [36, 27]

$S_{\rm NLO}=-2\pi\lim_{q^{2}\to
0}\hat{\Pi}(q^{2})=-\frac{N+2}{24\pi}\left(1+\log\left(\frac{M^{2}}{\mu^{2}}\right)\right)-16\pi
L_{10}\ .$ (10)

Assuming our result for $L_{10}$ can be used in the M6 model (for which $N=5$
in the above equation), we find for that model

$S_{\rm NLO}=0.8(2)\ ,$ (11)

where the error is dominated by the systematic error of $L_{10}$. In order to
arrive at this result we have assumed for simplicity that the 14 pseudo Nambu-
Goldstone bosons of the SU(5)/SO(5) coset have a common mass $M$ about the
size of the Higgs mass or somewhat larger. The uncertainty due to the actual
spread of (non-degenerate) masses is expected to be much smaller than the
already quoted error. Also, we assumed $F_{6}=1.1$ TeV, the lowest value
consistent with the commonly quoted upper bound $\xi\leq 0.1$ [4, 5, 6]. The
current experimental estimate is $S=-0.01(10)$, which implies a $1\sigma$
upper bound of 0.09. This yields an independent $1\sigma$ bound

$\xi\leq\frac{0.09}{0.8(2)}=0.11(3)\ ,$ (12)

which is compatible with the upper bound mentioned above.

##  4 Future prospects

We previously reported an acute problem with our prototype composite-Higgs
model [11, 37]. The problem has to do with the partially-composite top, which
should receive its mass via direct coupling to a hypercolor baryon with the
same Standard-Model quantum numbers as the top quark. This coupling is
facilitated by a 4-fermion operator, schematically, $tQqq$, where $t$ is the
top quark, $Q$ is a sextet hypercolor fermion, while $qq$ is a pair of
fundamental hypercolor fermions. The hypercolor singlet operator $Qqq$
generates a “chimera” baryon.

The 4-fermion coupling $G/\Lambda_{UV}^{2}$ (with dimensionless $G$) is
assumed to arise from physics at a new ultraviolet scale $\Lambda_{UV}$. In
order to meet flavor constraints, $\Lambda_{UV}$ should be much higher than
the hypercolor scale $\Lambda_{HC}$ [4, 5, 6]. We calculated the matrix
element of $Qqq$ between the vacuum and a chimera state, finding a very small
value. Demanding the hierarchy $\Lambda_{UV}\gg\Lambda_{HC}$, our result
implies that the top quark can only receive a mass which is smaller by several
orders of magnitude than its actual mass. Conversely, if we insist that the
top quark receive the correct mass, then $\Lambda_{UV}$ would have to be
smaller than $\Lambda_{HC}$. Either way, the prototype model is ruled out as a
realistic composite-Higgs model.

The remedy might come from enlarging the fermion content of the hypercolor
theory, driving it close to the conformal window, or even into it. Infrared
conformal theories, as well as walking theories, have anomalous dimensions
that (a) can be large, and (b) can stay roughly constant over many energy
decades. In particular, a large anomalous dimension for the chimera operator
$Qqq$ could enhance the 4-fermion coupling $G$ by several orders of magnitude,
eventually leading to a phenomenologically acceptable partially-composite top.

Figure 2:  Estimates for the location of the conformal window of the SU(4)
gauge theory with $N_{f}$ Dirac fermions in the fundamental representation and
$n_{f}$ Majorana fermions in the sextet representation. The uppermost line is
the limit of asymptotic freedom. The green band represents the analytical
estimate of Ref. [38] for the conformal window, while the dashed lines are
other analytical estimates of the bottom of the window. Black square: 2+2
model; Blue circle: M6 model; red diamond: M11 model; open circle: 4+4 model.
Adapted from Ref. [38].

Figure 2, adapted from Ref. [38], shows various analytical estimates for the
conformal window for the SU(4) gauge theory in the fundamental–sextet plane.
Our prototype “$2+2$ model” (with 2 Dirac fermions in each of the fundamental
and sextet representations) is indicated by the black square near the lower-
left corner. Doubling the number of fermions in each representation brings us
to the open circle, or “$4+4$ model.” One can see that the $2+2$ model lies
well below the sill of the conformal window according to all analytical
estimates. By contrast, the $4+4$ model would be inside the conformal window
according to some estimates, or slightly below it according to the others. In
addition, two models from the Ferretti-Karateev list: the M6 model mentioned
before, as well as the M11 model, can be reached from the $4+4$ model by
giving some of the fermions a large mass.

In summary, the $4+4$ model can be infrared conformal or near conformal
according to analytical estimates, making it a promising candidate for a
composite Higgs model. It would be interesting to study the $4+4$ model using
similar methods to those we have employed for the $2+2$ model.

Acknowledgements

Our calculations of staggered fermion propagators and currents were carried
out with code derived from version 7.8 of the publicly available code of the
MILC collaboration [39]. Computations for this work were carried out with
resources provided by the USQCD Collaboration, which is funded by the Office
of Science of the U.S. Department of Energy. This material is based upon work
supported by the U.S. Department of Energy, Office of Science, Office of High
Energy Physics, under Awards No. DE-SC0010005 (Colorado) and DE-SC0013682
(SFSU), and by the Israel Science Foundation under grant No. 491/17 (Tel
Aviv). Fermilab is operated by the Fermi Research Alliance, LLC under contract
No. DE-AC02-07CH11359 with the U.S. Department of Energy.

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