# Inscriptis - A Python-based HTML to text conversion library optimized for
knowledge extraction from the Web

Albert Weichselbraun

(Swiss Institute for Information Science
University of Applied Sciences of the Grisons
Pulvermühlestrasse 57, Chur, Switzerland)

##  1 Summary

Inscriptis provides a library, command line client and Web service for
converting HTML to plain text.

Its development has been triggered by the need to obtain accurate text
representations for knowledge extraction tasks that preserve the spatial
alignment of text without drawing upon heavyweight, browser-based solutions
such as Selenium [9]. In contrast to existing software packages such as
HTML2text [23], jusText [2] and Lynx [5], Inscriptis

  1. 1.

provides a layout-aware conversion of HTML that more closely resembles the
rendering obtained from standard Web browsers and, therefore, better preserves
the spatial arrangement of text elements. Inscriptis excels in terms of
conversion quality, since it correctly converts complex HTML constructs such
as nested tables and also interprets a subset of HTML (e.g., align, valign)
and CSS (e.g., display, white-space, margin-top, vertical-align, etc.)
attributes that determine the text alignment.

  2. 2.

supports annotation rules, i.e., user-provided mappings that allow for
annotating the extracted text based on structural and semantic information
encoded in HTML tags and attributes used for controlling structure and layout
in the original HTML document.

These unique features ensure that downstream knowledge extraction components
can operate on accurate text representations, and may even use information on
the semantics and structure of the original HTML document, if annotation
support has been enabled.

##  2 Statement of need

Research in a growing number of scientific disciplines relies upon Web
content. Li et al. [12], for instance, studied the impact of company-specific
News coverage on stock prices, in medicine and pharmacovigilance social media
listening plays an important role in gathering insights into patient needs and
the monitoring of adverse drug effects [4], and communication sciences analyze
media coverage to obtain information on the perception and framing of issues
as well as on the rise and fall of topics within News and social media [21,
26].

Computer science focuses on analyzing content by applying knowledge extraction
techniques such as entity recognition [8] to automatically identify entities
(e.g., persons, organizations, locations, products, etc.) within text
documents, entity linking [6] to link these entities to knowledge bases such
as Wikidata and DBPedia, and sentiment analysis to automatically assess
sentiment polarity (i.e., positive versus negative coverage) and emotions
expressed towards these entities [24].

Most knowledge extraction methods operate on text and, therefore, require an
accurate conversion of HTML content which also preserves the spatial alignment
between text elements. This is particularly true for methods drawing upon
algorithms which directly or indirectly leverage information on the proximity
between terms, such as word embeddings [13, 16], language models [18],
sentiment analysis which often also considers the distance between target and
sentiment terms, and automatic keyword and phrase extraction techniques.

Despite this need from within the research community, many standard HTML to
text conversion techniques are not layout aware, yielding text representations
that fail to preserve the text’s spatial properties, as illustrated below:

Figure 1: Text representation of a table from Wikipedia computed by Inscriptis
(left) and Lynx (right). Lynx fails to correctly interpret the table and,
therefore, does not properly align the temperature values.

Consequently, even popular resources extensively used in literature suffer
from such shortcomings. The text representations provided with the Common
Crawl corpus111https://commoncrawl.org/, for instance, have been generated
with a custom utility [10] which at the time of writing did not consider any
layout information. Datasets such as CCAligned [7] multilingual C4 which has
been used for training the mT5 language model [27], and OSCAR [22] are based
on subsets of the Common Crawl corpus [3].

Even worse, some tutorials suggest the use of software libraries such as
Beautiful Soup [19], lxml [1] and Cheerio [14] for converting HTML. Since
these libraries have been designed with a different use case in mind, they are
only well-suited for scraping textual content. Once they encounter HTML
constructs such as lists and tables, these libraries are likely to return
artifacts (e.g., concatenated words), since they do not interpret HTML
semantics. The creators of the Cheerio library even warn their users, by
explicitly stating that it is not well-suited for emulating Web browsers.

Specialized conversion tools such as HTML2Text perform considerably better but
often fail for more complex Web pages. Researchers sometimes even draw upon
text-based Web browsers such as Lynx to obtain more accurate representations
of HTML pages. These tools are complemented by content extraction software
such as jusText [2], dragnet [17], TextSweeper [11] and boilerpy3 [20] which
do not consider the page layout but rather aim at extracting the relevant
content only, and approaches that are optimized for certain kinds of Web pages
like Harvest [25] for Web forums.

Inscriptis, in contrast, not only correctly renders more complex websites but
also offers the option to preserve parts of the original HTML document’s
semantics (e.g., information on headings, emphasized text, tables, etc.) by
complementing the extracted text with annotations obtained from the document.
Figure 2 provides an example of annotations extracted from a Wikipedia page.
These annotations can be useful for

  * •

providing downstream knowledge extraction components with additional
information that may be leveraged to improve their respective performance.
Text summarization techniques, for instance, can put a stronger emphasis on
paragraphs that contain bold and italic text, and sentiment analysis may
consider this information in addition to textual clues such as uppercase text.

  * •

assisting manual document annotation processes (e.g., for qualitative analysis
or gold standard creation). Inscriptis supports multiple export formats such
as XML, annotated HTML and the JSONL format that is used by the open source
annotation tool doccano222Please note that doccano currently does not support
overlapping annotations and, therefore, cannot import files containing
overlapping annotations. [15]. Support for further annotation formats can be
easily added by implementing custom annotation post-processors.

  * •

enabling the use of Inscriptis for tasks such as content extraction (i.e.,
extract task-specific relevant content from a Web page) which rely on
information on the HTML document’s structure.

Figure 2: Annotations extracted from the Wikipedia entry for Chur that have
been exported to HTML using the \--postprocessor html command line option.

In conclusion, Inscriptis provides knowledge extraction components with high
quality text representations of HTML documents. Since its first public release
in March 2016, Inscriptis has been downloaded over 135,000 times from the
Python Package Index (PyPI)333Source: https://pepy.tech/project/inscriptis,
has proven its capabilities in national and European research projects, and
has been integrated into commercial products such as the webLyzard Web
Intelligence and Visual Analytics Platform.

##  3 Mentions

The following research projects use Inscriptis within their knowledge
extraction pipelines:

  * •

CareerCoach: “Automatic Knowledge Extraction and Recommender Systems for
Personalized Re- and Upskilling suggestions” funded by Innosuisse.

  * •

Job Cockpit: “Web analytics, data enrichment and predictive analysis for
improved recruitment and career management processes” funded by Innosuisse

  * •

EPOCH project funded by the Austrian Federal Ministry for Climate Action,
Environment, Energy, Mobility and Technology (BMK) via the ICT of the Future
Program.

  * •

MedMon: “Monitoring of Internet Resources for Pharmaceutical Research and
Development” funded by Innosuisse.

  * •

ReTV project funded by the European Union’s Horizon 2020 Research and
Innovation Programme.

##  4 Acknowledgements

Work on Inscriptis has been conducted within the MedMon, Job Cockpit and
CareerCoach projects funded by Innosuisse.

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# Genuine N-partite entanglement in Schwarzschild-de Sitter black hole
spacetime

Shu-Min Wu111Email<EMAIL_ADDRESS>Xiao-Wei Teng, Xiao-Li Huang222Email:
<EMAIL_ADDRESS>Jianbo Lu333Email<EMAIL_ADDRESS>Department
of Physics, Liaoning Normal University, Dalian 116029, China

###### Abstract

Complex quantum information tasks in a gravitational background require
multipartite entanglement for effective processing. Therefore, it is necessary
to investigate the properties of multipartite entanglement in a relativistic
setting. In this paper, we study genuine N-partite entanglement of massless
Dirac fields in the Schwarzschild-de Sitter (SdS) spacetime, characterized by
the presence of a black hole event horizon (BEH) and a cosmological event
horizon (CEH). We obtain the general analytical expression of genuine
N-partite entanglement shared by $n$ observers near BEH and $m$ ($n+m=N$)
observers near CEH. It is shown that genuine N-partite entanglement
monotonically decreases with the decrease of the mass of the black hole,
suggesting that the Hawking effect of the black hole destroys quantum
entanglement. It is interesting to note that genuine N-partite entanglement is
a non-monotonic function of the cosmological constant, meaning that the
Hawking effect of the expanding universe can enhance quantum entanglement.
This result contrasts with multipartite entanglement in single-event horizon
spacetime, offering a new perspective on the Hawking effect in multi-event
horizon spacetime.

###### pacs:

04.70.Dy, 03.65.Ud,04.62.+v

##  I Introduction

Quantum entanglement plays an important role in numerous quantum information
processing tasks, including quantum cryptography, quantum teleportation, and
dense coding L1 ; L2 ; L3 ; L4 . Unlike the usual multipartite entangled
state, a genuine multipartite entangled state cannot be separated into any
bipartite partitions. Genuine multipartite entanglement offers advantages over
usual entanglement in the key resources for measurement-based quantum
computing and high-precision metrology L5 ; L6 . Understanding genuine
multipartite entanglement in a relativistic framework is crucial, especially
considering the inevitable influence of gravity on quantum entanglement in
real-world environments. It is important to note that investigations of
genuine multipartite entanglement near the event horizon of the black hole
have been mainly limited to asymptotically flat spacetime L8 ; L9 ; L10 ; L11
; L12 ; L13 ; L14 ; L15 ; L16 ; L17 ; L18 ; L19 ; L20 ; L21 . To clarify,
genuine multipartite entanglement has been mainly studied in the single-event
horizon spacetime. The research papers have illustrated that the relativistic
effects in this spacetime lead to the degradation of genuine multipartite
entanglement.

The de Sitter solution is widely recognized as the most straightforward
solution derived from Einstein’s field equations with a nonvanishing
cosmological constant L22 ; L23 ; L24 ; L25 ; LL25 . The study of phenomena in
asymptotically de Sitter spacetimes is imperative and holds significant
interest, particularly in light of experimental evidence indicating the
accelerating expansion of our universe L26 ; L27 . In reality, the black hole
is asymptotically de Sitter, rather than asymptotically flat. A static,
chargeless black hole is associated with the Schwarzschild-de Sitter (SdS)
spacetime, characterized by the mass $M$ of the black hole and the
cosmological constant $\Lambda$. Therefore, the SdS spacetime features both a
black hole event horizon (BEH) and a cosmological event horizon (CEH),
introducing two-temperature thermodynamics distinct from those observed in
single-event horizon spacetime L28 ; L29 ; L30 ; L31 . In comparison to
single-event horizon spacetime, studying quantum information in multi-event
horizon spacetime is more realistic, especially the exploration of
multipartite entanglement, which has been a gap in the research. In addition,
as relativistic quantum information tasks grow in complexity, the utilization
of multipartite entanglement becomes essential for their processing. Hence,
studying the relativistic effects of the SdS spacetime on genuine N-partite
entanglement is one of the motivations for our work. Another motivation for
our work is better to understand the multi-event horizon SdS spacetime through
genuine N-partite entanglement.

In this paper, we study the properties of genuine N-partite entanglement of
Dirac fields in SdS spacetime endowed with the BEH and the CEH. Our model
comprises $N$ modes: (i) the $n$ ($n<N$) modes located at the BEH; (ii) the
$m$ ($n+m=N$) modes situated at the CEH. We will derive the analytical
expression for genuine N-partite entanglement in multi-event horizon
spacetime. We aim to investigate how the Hawking effect of the black hole and
the Hawking effect of the expanding universe influence genuine N-partite
entanglement. Additionally, we will investigate how genuine N-partite
entanglement depends on $n$ and $m$ in the context of the multi-event horizon
spacetime. As we all know, the gravitational effect of the single-event
horizon spacetime destroys genuine N-partite entanglement L9 ; L10 ; L11 ; L12
; L13 ; L14 ; L15 ; L16 ; L17 ; L18 ; L19 ; L20 ; L21 . An intriguing question
arises: will the gravitational effect of the multi-event horizon spacetime
increase genuine N-partite entanglement?

The structure of the paper is as follows. In Sec. II, we describe the
quantization of Dirac field in SdS spacetime. In Sec. III, we study the
influence of the Hawking effect on genuine N-partite entanglement in multi-
event horizon spacetime. The last section is devoted to the summary.

##  II Quantization of Dirac field in SdS spacetime

The SdS spacetime metric is the unique solution to Einstein’s field equations,
including a positive cosmological constant $\Lambda$ in (3+1)-spacetime
dimensions L28 . The metric of the SdS spacetime can be given as

$\displaystyle ds^{2}=-\bigg{(}1-\frac{2M}{r}-\frac{\Lambda
r^{2}}{3}\bigg{)}dt^{2}+\bigg{(}1-\frac{2M}{r}-\frac{\Lambda
r^{2}}{3}\bigg{)}^{-1}dr^{2}+r^{2}\big{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\big{)}.$ (1)

We shall now introduce the horizon structure of the SdS spacetime, which is
dependent on the cosmological constant $\Lambda$. For a critical value of
$\Lambda_{\rm crit}=1/(9M)$, the event horizon of the SdS spacetime does not
exist and the corresponding solution is represented as the naked singularity.
In the range $0<\Lambda<\Lambda_{\rm crit}$, the SdS spacetime has the black
hole event horizon (BEH), the cosmological event horizon (CEH), and the
unphysical event horizon for $f(r)=1-\frac{2M}{r}-\frac{\Lambda r^{2}}{3}=0$
L32 . In this paper, we only consider the scope $0<\Lambda<\Lambda_{\rm crit}$
along with the lapse function $f(r)$ that takes the form as

$\displaystyle f(r)=\frac{\Lambda}{3r}(r_{H}-r)(r-r_{C})(r+r_{H}+r_{C}).$ (2)

Here, the expressions for the $r_{H}$ (BEH) and $r_{C}$ (CEH) in terms of the
mass of the black hole and the cosmological constant can be written as

$\displaystyle
r_{H}=\frac{2}{\sqrt{\Lambda}}\cos\bigg{[}\frac{\pi+\arccos(3M\sqrt{\Lambda})}{3}\bigg{]},\quad
r_{C}=\frac{2}{\sqrt{\Lambda}}\cos\bigg{[}\frac{\arccos(3M\sqrt{\Lambda})-\pi}{3}\bigg{]}.$
(3)

The surface gravities of the black hole and the expanding universe can both be
denoted as

$\displaystyle\kappa_{H}=\frac{\Lambda(2r_{H}+r_{C})(r_{C}-r_{H})}{6r_{H}},\quad-\kappa_{C}=\frac{\Lambda(2r_{C}+r_{H})(r_{H}-r_{C})}{6r_{C}}.$
(4)

From the equation above, it becomes apparent that the surface gravity of the
expanding universe manifests as negative, attributed to the repulsive effects
induced by $\Lambda>0$. Because of $r_{H}<r_{C}$, we obtain
$\kappa_{H}>\kappa_{C}$, showing that the Hawking temperature of the expanding
universe $T_{C}=\frac{\kappa_{C}}{2\pi}$ is smaller than the Hawking
temperature of the black hole $T_{H}=\frac{\kappa_{H}}{2\pi}$ L33 ; L34 ; L35
.

To obtain the metric in the Kruskal coordinates, we introduce the tortoise
coordinate, undergoing transforms as $\mu=t-r_{\star}$ and $\nu=t+r_{\star}$,
wherein the tortoise coordinate is denoted by

$\displaystyle
r_{\star}=\frac{1}{2\kappa_{H}}\ln\big{|}\frac{r}{r_{H}}-1\big{|}-\frac{1}{2\kappa_{C}}\ln\big{|}1-\frac{r}{r_{C}}\big{|}+\frac{1}{2\kappa_{U}}\ln\big{|}\frac{r}{r_{U}}-1\big{|},$
(5)

with $r_{U}=-(r_{H}+r_{C})$ ZL35 ; ZZL35 . Here, $\kappa_{U}$ is the surface
gravity of the unphysical horizon $r_{U}$. We need two Kruskal coordinate
patches to get the non-singular coordinate mapping for the entire SdS
spacetime manifold through analytical continuation. The Kruskal coordinates
are found to be

$\displaystyle\bar{\mu}_{H}=-\frac{1}{\kappa_{H}}e^{-\kappa_{H}\mu},\quad\bar{\nu}_{H}=\frac{1}{\kappa_{H}}e^{\kappa_{H}\nu},\quad\bar{\mu}_{C}=\frac{1}{\kappa_{C}}e^{\kappa_{C}\mu},\quad\bar{\nu}_{C}=-\frac{1}{\kappa_{C}}e^{-\kappa_{C}\nu}.$
(6)

Finally, the BEH and CEH description of the metric in terms of the Kruskal
coordinate can be expressed as

$\displaystyle
ds^{2}=-\frac{2M}{r}\big{|}1-\frac{r}{r_{C}}\big{|}^{1+\frac{\kappa_{H}}{\kappa_{C}}}\big{(}1+\frac{r}{r_{H}+r_{C}}\big{)}^{1-\frac{\kappa_{H}}{\kappa_{U}}}d\bar{\mu}_{H}d\bar{\nu}_{H}+r^{2}\Omega_{2}^{2},$
(7) $\displaystyle
ds^{2}=-\frac{2M}{r}\big{|}\frac{r}{r_{H}}-1\big{|}^{1+\frac{\kappa_{C}}{\kappa_{H}}}\big{(}1+\frac{r}{r_{H}+r_{C}}\big{)}^{1+\frac{\kappa_{C}}{\kappa_{U}}}d\bar{\mu}_{C}d\bar{\nu}_{C}+r^{2}\Omega_{2}^{2}.$
(8)

Figure 1:  The SdS spacetime with thermal opaque membrane.

Upon review, it was discovered that the SdS spacetime has two physical event
horizons associated with different local temperatures, meaning that they
cannot reach thermal equilibrium. In order to simplify the analysis, it is
crucial to ensure that the system is in thermal equilibrium. The thermal
opaque membrane serves this purpose. By employing it, it becomes possible to
analyze one horizon with another as the boundary in the multi-event horizon
SdS spacetime L36 ; L37 . Therefore, the thermally opaque membrane divides
region $C$ into two sub-regions, namely $A$ and $B$ ($C=A\cup B$) in Fig.1. In
our model, we consider that the $n$ observers located at the BEH can detect
the Hawking radiation at temperature $T_{H}$, while the $m$ observers situated
at the CEH can detect the Hawking radiation at temperature $T_{C}$.

The massless Dirac equation can be specifically expressed in the following
form L38

$\displaystyle[\gamma^{a}e_{a}{}^{\mu}(\partial_{\mu}+\Gamma_{\mu})]\Phi=0,$
(9)

where $\gamma^{a}$ are the Dirac matrices and the four-vectors $e_{a}{}^{\mu}$
is the inverse of the tetrad $e^{a}{}_{\mu}$. Let’s first consider the sub-
region $A$ and the causally disconnected region $L$, which faces the BEH in
Fig.1. The field quantization in the SdS spacetime can be performed in a
similar way to the Unruh effect L38 ; L39 ; L40 ; L41 , and we will not
discuss it in detail here. Therefore, one can employ the black hole mode and
the Kruskal mode for the quantization of the Dirac field, respectively, and
then get the Bogoliubov transformations of the operators in SdS and Kruskal
coordinates L38 ; L42 . According to Bogoliubov transformations, the
expressions for the Kruskal vacuum state and the excited state in the black
hole spacetime are found to be

$\displaystyle|0_{K_{H}}\rangle$ $\displaystyle=$ $\displaystyle\cos
r|0_{A},0_{L}\rangle+\sin r|1_{A},1_{L}\rangle,$
$\displaystyle|1_{K_{H}}\rangle$ $\displaystyle=$
$\displaystyle|1_{A},0_{L}\rangle,$ (10)

with $\cos r=\frac{1}{\sqrt{e^{-\frac{\omega}{T_{H}}}+1}}$. Here,
$|n_{A}\rangle$ and $|n_{L}\rangle$ denote the number states corresponding to
the fermion outside the event horizon and the antifermion inside the event
horizon of the black hole, respectively. Similarly, the expressions of the
Kruskal vacuum state and the excited state in the expanding universe can be
shown as

$\displaystyle|0_{K_{C}}\rangle$ $\displaystyle=$ $\displaystyle\cos
w|0_{B},0_{R}\rangle+\sin w|1_{B},1_{R}\rangle,$
$\displaystyle|1_{K_{C}}\rangle$ $\displaystyle=$
$\displaystyle|1_{B},0_{R}\rangle,$ (11)

with $\cos w=\frac{1}{\sqrt{e^{-\frac{\omega}{T_{C}}}+1}}$. Due to causal
disconnection, the observer in the sub-region $A$ or $B$ cannot detect the
modes of regions $L$ and $R$.

##  III Genuine N-partite entanglement in SdS spacetime

If a state of the N-partite system is not biseparable, it is named a genuinely
N-partite entangled state. Here, we introduce the concurrence as a measure for
genuine N-partite entanglement. The density matrix for the X-state of the
N-partite system in the Hilbert-space orthonormal bases
$\\{|0,0,...,0\rangle,|0,0,...,1\rangle,...,|1,1,...,1\rangle\\}$ can be
expressed as

$\displaystyle\rho_{X}=\left(\\!\\!\begin{array}[]{cccccccc}\mathcal{M}_{1}&&&&&&&\mathcal{C}_{1}\\\
&\mathcal{M}_{2}&&&&&\mathcal{C}_{2}&\\\ &&\ddots&&&\mathinner{\mskip
1.0mu\raise 1.0pt\vbox{\kern 7.0pt\hbox{.}}\mskip 2.0mu\raise
4.0pt\hbox{.}\mskip 2.0mu\raise 7.0pt\hbox{.}\mskip 1.0mu}&&\\\
&&&\mathcal{M}_{y}&\mathcal{C}_{y}&&&\\\
&&&\mathcal{C}_{y}^{*}&\mathcal{N}_{y}&&&\\\ &&\mathinner{\mskip 1.0mu\raise
1.0pt\vbox{\kern 7.0pt\hbox{.}}\mskip 2.0mu\raise 4.0pt\hbox{.}\mskip
2.0mu\raise 7.0pt\hbox{.}\mskip 1.0mu}&&&\ddots&&\\\
&\mathcal{C}_{2}^{*}&&&&&\mathcal{N}_{2}&\\\
\mathcal{C}_{1}^{*}&&&&&&&\mathcal{N}_{1}\end{array}\\!\\!\right),$ (20)

with $y=2^{N-1}$. The conditions $\sum_{i}(\mathcal{M}_{i}+\mathcal{N}_{i})=1$
and $|\mathcal{C}_{i}|\leq\sqrt{\mathcal{M}_{i}\mathcal{N}_{i}}$ show that the
X-state $\rho_{X}$ is normalized and positive. Genuine N-partite concurrence
can be denoted as

$\displaystyle
C(\rho_{X})=2\max\\{0,|\mathcal{C}_{i}|-\mu_{i}\\},i=1,\ldots,y,$ (21)

where $\mu_{i}=\sum_{j\neq i}^{y}\sqrt{\mathcal{M}_{j}\mathcal{N}_{j}}$ L43 .

In this paper, we initially consider an N-partite Greenberger-Horne-Zeilinger
(GHZ) entangled state with $n$ Kruskal modes $\kappa_{H}$ and $m$ Kruskal
modes $\kappa_{C}$,

$\displaystyle|\psi\rangle_{1,\ldots,N}$ $\displaystyle=$
$\displaystyle\alpha|0_{\kappa_{H}}^{1},0_{\kappa_{H}}^{2},...,0_{\kappa_{H}}^{n},0_{\kappa_{C}}^{1},0_{\kappa_{C}}^{2},...,0_{\kappa_{C}}^{m}\rangle$
$\displaystyle+$
$\displaystyle\sqrt{1-\alpha^{2}}|1_{\kappa_{H}}^{1},1_{\kappa_{H}}^{2},...,1_{\kappa_{H}}^{n},1_{\kappa_{C}}^{1},1_{\kappa_{C}}^{2},...,1_{\kappa_{C}}^{m}\rangle.$

Here, $n$ ($0<n<N$) and $m$ ($0<m<N$) satisfy the relationship $n+m=N$. To the
Kruskal observers, the quantum system consists of $N$ modes. Next, we let the
$n$ modes be located at the BEH in the sub-region $A$ of $C$, and the $m$
modes are located at the CEH in the sub-region $B$ of $C$. Due to the Hawking
effects of the SdS spacetime, the additional $n$ modes and $m$ modes appear in
the region $L$ and the region $R$ in Fig.1, respectively. Using Eqs.(II) and
(II), we can rewrite the initial state of Eq.(III) as

$\displaystyle|\psi\rangle_{1,\ldots,2N}$ $\displaystyle=$
$\displaystyle\alpha\bigg{[}\bigotimes_{i=1}^{n}(\cos
r|0^{i}_{A},0^{i}_{L}\rangle+\sin
r|1^{i}_{A},1^{i}_{L}\rangle)\bigotimes_{j=1}^{m}(\cos
w|0^{j}_{B},0^{j}_{R}\rangle+\sin w|1^{j}_{B},1^{j}_{R}\rangle)\bigg{]}$
$\displaystyle+$
$\displaystyle\sqrt{1-\alpha^{2}}\bigg{[}\bigotimes_{x=1}^{n}(|1^{x}_{A},0^{x}_{L}\rangle)\bigotimes_{y=1}^{m}(|1^{y}_{B},0^{y}_{R}\rangle)\bigg{]}.$

Since the sub-regions $A$ and $B$ are causally disconnected from the sub-
regions $L$ and $R$, we should take the trace over physically inaccessible
modes in the sub-regions L and R and then obtain the density operator
$\rho_{N}$ as

$\displaystyle\rho_{N}=\rho_{\bm{A}}+\rho_{\bm{X}}+\rho^{\dagger}_{\bm{X}}+\rho_{\bm{B}},$
(24)

with

$\displaystyle\rho_{\bm{A}}$ $\displaystyle=$
$\displaystyle\alpha^{2}\bigg{\\{}\bigg{[}\cos^{2n}r\bigotimes_{i=1}^{n}(|0\rangle^{i}_{A}\langle
0|)\bigg{]}\bigg{[}\cos^{2m}w\bigotimes_{j=1}^{m}(|0\rangle^{j}_{B}\langle
0|)\bigg{]}+\bigg{[}\cos^{2n}r\bigotimes_{i=1}^{n}(|0\rangle^{i}_{A}\langle
0|)\bigg{]}$ $\displaystyle\times$
$\displaystyle\bigg{[}\cos^{2(m-1)}w\sin^{2}w\bigotimes_{j=1}^{m-1}(|0\rangle^{j}_{B}\langle
0|)|1\rangle^{m}_{B}\langle 1|\bigg{]}+...$ $\displaystyle+$
$\displaystyle\bigg{[}\cos^{2}r\sin^{2(n-1)}r|0\rangle^{1}_{A}\langle
0|\bigotimes_{i=2}^{n}(|1\rangle^{i}_{A}\langle
1|)\bigg{]}\bigg{[}\cos^{2}w\sin^{2(m-1)}w\bigotimes_{j=1}^{m-1}(|1\rangle^{j}_{B}\langle
1|)|0\rangle^{m}_{B}\langle 0|\bigg{]}$ $\displaystyle+$
$\displaystyle\bigg{[}\cos^{2}r\sin^{2(n-1)}r|0\rangle^{1}_{A}\langle
0|\bigotimes_{i=2}^{n}(|1\rangle^{i}_{A}\langle
1|)\bigg{]}\bigg{[}\sin^{2m}w\bigotimes_{j=1}^{m}(|1\rangle^{j}_{B}\langle
1|)\bigg{]}\bigg{]}\bigg{\\}},$
$\displaystyle\rho_{\bm{X}}=\alpha\sqrt{1-\alpha^{2}}\bigg{[}\cos^{n}r\bigotimes_{i=1}^{n}(|0\rangle^{i}_{A}\langle
1|)\bigg{]}\bigg{[}\cos^{m}w\bigotimes_{j=1}^{m}(|0\rangle^{j}_{B}\langle
1|)\bigg{]},$

and

$\displaystyle\rho_{\bm{B}}$ $\displaystyle=$
$\displaystyle\alpha^{2}\bigg{\\{}\bigg{[}\cos^{2(n-1)}r\sin^{2}r|1\rangle^{1}_{A}\langle
1|\bigotimes_{i=2}^{n}(|0\rangle^{i}_{A}\langle
0|)\bigg{]}\bigg{[}\cos^{2m}w\bigotimes_{j=1}^{m}(|0\rangle^{j}_{B}\langle
0|)\bigg{]}$ $\displaystyle+$
$\displaystyle\bigg{[}\cos^{2(n-1)}r\sin^{2}r|1\rangle^{1}_{A}\langle
1|\bigotimes_{i=2}^{n}(|0\rangle^{i}_{A}\langle
0|)\bigg{]}\bigg{[}\cos^{2(m-1)}w\sin^{2}w\bigotimes_{j=1}^{m-1}(|0\rangle^{j}_{B}\langle
0|)|1\rangle^{m}_{B}\langle 1|\bigg{]}$ $\displaystyle+$
$\displaystyle...+\bigg{[}\sin^{2n}r\bigotimes_{i=1}^{n}(|1\rangle^{i}_{A}\langle
1|)\bigg{]}\bigg{[}\cos^{2}w\sin^{2(m-1)}w\bigotimes_{j=1}^{m-1}(|1\rangle^{j}_{B}\langle
1|)|0\rangle^{m}_{B}\langle 0|\bigg{]}\bigg{\\}}$ $\displaystyle+$
$\displaystyle\bigg{[}\alpha^{2}\sin^{2n}r\sin^{2m}w+(1-\alpha^{2})\bigg{]}\bigg{[}\bigotimes_{i=1}^{n}(|1\rangle^{i}_{A}\langle
1|)\bigg{]}\bigg{[}\bigotimes_{j=1}^{m}(|1\rangle^{j}_{B}\langle 1|)\bigg{]},$

which we write in matrix form as

$\displaystyle\rho_{N}=\left(\\!\\!\begin{array}[]{cc}\mathcal{M}_{\bm{A}}&\mathcal{M}_{\bm{X}}\\\
\mathcal{M}_{\bm{X}}^{T}&\mathcal{M}_{\bm{B}}\\\ \end{array}\\!\\!\right),$
(27)

in the $2^{N}$ basis

$\displaystyle\\{|0^{1}_{A},...,0^{n}_{A},0^{1}_{B},...0^{m}_{B}\rangle,|0^{1}_{A},...,0^{n}_{A},0^{1}_{B},...0^{m-1}_{B},1^{m}_{B}\rangle,...,|1^{1}_{A},...,1^{n}_{A},1^{1}_{B},...1^{m-1}_{B},0^{m}_{B}\rangle,$
$\displaystyle|1^{1}_{A},...,1^{n}_{A},1^{1}_{B},...1^{m}_{B}\rangle\\}.$

The sub-matrixes $\mathcal{M}_{\bm{A}}$, $\mathcal{M}_{\bm{X}}$, and
$\mathcal{M}_{\bm{B}}$ are detailed in Appendix A.

Figure 2: Genuine N-partite entanglement $C(\rho_{N})$ as a function of the
mass $M$ of the black hole for different $n$ and $m$, where
$\omega=\Lambda=1$.

Employing Eqs.(21) and (27), we obtain the genuine N-partite entanglement
measured by the concurrence as

$\displaystyle
C(\rho_{N})=2\cos^{n}r\cos^{m}w\max\bigg{\\{}0,\alpha\sqrt{1-\alpha^{2}}-(2^{N-1}-1)\alpha^{2}\sin^{n}r\sin^{m}w\bigg{\\}}.$
(28)

From Eq.(28), it is easy to see that genuine N-partite entanglement depends
not only on the initial parameters $\alpha$, $n$, and $m$, but also on the
mass $M$ of the black hole and the cosmological constant $\Lambda$.

In Fig.2, we plot genuine N-partite entanglement $C(\rho_{N})$ as a function
of the mass $M$ of the black hole for different $n$ and $m$. From Fig.2, we
can observe that genuine N-partite entanglement decreases monotonically with
the decrease of the mass $M$ of the black hole, meaning that the Hawking
effect of the black hole degenerates quantum entanglement. Note that genuine
N-partite entanglement recovers to initial value
“$2\alpha\sqrt{1-\alpha^{2}}$” at the Nariai limit. This is because the two
event horizons do not exist in this limit. We find that genuine N-partite
entanglement monotonically decreases with increasing $n$, while it exhibits
non-monotonic changes with increasing $m$. Based on the Hawking temperature of
the black hole $T_{H}$ being greater than the Hawking temperature of the
expanding universe $T_{C}$, it can be concluded that genuine N-partite
entanglement increases with the increase of $m$ for a fixed initial total
number of particles $N$. In other words, genuine N-partite entanglement
monotonically decreases as $n$ increases for a fixed $N$. This conclusion is
also supported by Fig.2 (c) and (d).

Figure 3: Genuine N-partite entanglement $C(\rho_{N})$ and the Hawking
temperature $T_{C}$ of CEH as functions of the cosmological constant $\Lambda$
for different values of $n$ and $m$, where $M=0.033$ and $\omega=1$.

Fig.3 (a)-(c) shows how the cosmological constant $\Lambda$ influences genuine
N-partite entanglement $C(\rho_{N})$. From Fig.3 (d), we see that the Hawking
temperature $T_{C}$ of CEH changes non-monotonically with $\Lambda$. We find
that the Hawking effect of the expanding universe can enhance genuine
N-partite entanglement, while the Hawking effect of the black hole only
destroys genuine N-partite entanglement in the multi-event horizon spacetime.
This results are different from the property of multipartite entanglement in
the single-event horizon spacetime L9 ; L10 ; L11 ; L12 ; L13 ; L14 ; L15 ;
L16 ; L17 ; L18 ; L19 ; L20 ; L21 . Fig.3 (a)-(c) again shows that genuine
N-partite entanglement is a decreasing function with $n$ and a non-monotonic
function with $m$.

##  IV  Conclutions

In this paper, we have studied the effect of the Hawking effect of the
Schwarzschild-de Sitter (SdS) spacetime on genuine N-partite entanglement of
massless Dirac fields shared by $n$ observers near BEH and $m$ ($n+m=N$)
observers near CEH. We obtain the general analytical expression of genuine
N-partite entanglement for any $n$ and $m$ in the multi-event horizon
spacetime. We find that the Hawking effect of the black hole can only
degenerate genuine N-partite entanglement, while the Hawking effect of the
expanding universe can enhance it. However, the Hawking effect of the single-
event horizon spacetime destroys multipartite entanglement L9 ; L10 ; L11 ;
L12 ; L13 ; L14 ; L15 ; L16 ; L17 ; L18 ; L19 ; L20 ; L21 . This finding can
contribute to a better understanding of the Hawking effect in multi-event
horizon spacetime. This is because the Hawking effect of the black holes and
the Hawking effect of the expanding universe have different influences on
genuine N-partite entanglement. Since the Hawking temperature of the black
hole is bigger than the Hawking temperature of the expanding universe, genuine
N-partite entanglement increases with the increase of $m$ for a fixed initial
parameter $N$. These conclusions demonstrate the observer-dependent nature of
genuine N-partite entanglement in the multi-event horizon spacetime and guide
multipartite entanglement to deal with relativistic quantum information tasks.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 12205133, 12175095, and 12075050), LJKQZ20222315 and
JYTMS20231051, and LiaoNing Revitalization Talents Program (XLYC2007047).

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##  Appendix A Sub-matrixes $\mathcal{M}_{\bm{A}}$, $\mathcal{M}_{\bm{X}}$,
and $\mathcal{M}_{\bm{B}}$

By analyzing Eqs.(24) and (27), we find that the matrix $\rho_{N}$ is
$2^{N}\times 2^{N}$ dimensions, and the sub-matrixes $\mathcal{M}_{\bm{A}}$,
$\mathcal{M}_{\bm{X}}$, and $\mathcal{M}_{\bm{B}}$ are $2^{N-1}\times 2^{N-1}$
dimensions. First, the sub-density operator $\rho_{\bm{A}}$ corresponds to
sub-matrix $\mathcal{M}_{\bm{A}}$ in the basis

$\displaystyle\\{|0^{1}_{A},...,0^{n}_{A},0^{1}_{B},...0^{m}_{B}\rangle,|0^{1}_{A},...,0^{n}_{A},0^{1}_{B},...0^{m-1}_{B},1^{m}_{B}\rangle,...,|0^{1}_{A},1^{2}_{A},...,1^{n}_{A},1^{1}_{B},...1^{m-1}_{B},0^{m}_{B}\rangle,$
$\displaystyle|0^{1}_{A},1^{2}_{A},...,1^{n}_{A},1^{1}_{B},...1^{m}_{B}\rangle\\},$

where the base corresponding to the element
$\alpha^{2}\cos^{2(n-i)}r\sin^{2i}r\cos^{2(m-j)}w\sin^{2j}w$ include $i$
“$1_{A}$” and $j$ “$1_{B}$”,

$\displaystyle|0^{1}_{A},...,0^{n}_{A},0^{1}_{B},...0^{m}_{B}\rangle\langle
0^{1}_{A},...,0^{n}_{A},0^{1}_{B},...0^{m}_{B}|:\alpha^{2}\cos^{2n}r\cos^{2m}w,$
$\displaystyle|0^{1}_{A},...,0^{n}_{A},0^{1}_{B},...0^{m-1}1^{m}_{B}\rangle\langle
0^{1}_{A},...,0^{n}_{A},0^{1}_{B},...0^{m-1}1^{m}_{B}|:\alpha^{2}\cos^{2n}r\cos^{2(m-1)}w\sin^{2}w,...,$
$\displaystyle|0^{1}_{A},1^{2}_{A},...,1^{n}_{A},1^{1}_{B},...1^{m-1}_{B},0^{m}_{B}\rangle\langle
0^{1}_{A},1^{2}_{A},...,1^{n}_{A},1^{1}_{B},...1^{m-1}_{B},0^{m}_{B}|:\alpha^{2}\cos^{2}r\sin^{2(n-1)}r\cos^{2}w\times$
$\displaystyle\sin^{2(m-1)}w,|0^{1}_{A},1^{2}_{A},...,1^{n}_{A},1^{1}_{B},...1^{m}_{B}\rangle\langle
0^{1}_{A},1^{2}_{A},...,1^{n}_{A},1^{1}_{B},...1^{m}_{B}|:\alpha^{2}\cos^{2}r\sin^{2(n-1)}r\sin^{2m}w.$

Therefore, the sub-matrix $\mathcal{M}_{\bm{A}}$ can be written as

$\displaystyle\mathcal{M}_{\bm{A}}=\alpha^{2}\left(\\!\\!\begin{array}[]{cccc}\cos^{2n}r\cos^{2m}w&&&\\\
&\cos^{2n}r\cos^{2(m-1)}w\sin^{2}w&&\\\ &&\ddots&\\\
&&&\cos^{2}r\sin^{2(n-1)}r\sin^{2m}w\\\ \end{array}\\!\\!\right).$ (33)

Second, the sub-density operator $\rho_{\bm{X}}$ corresponds to sub-matrix
$\mathcal{M}_{\bm{X}}$ that can be easily written as

$\displaystyle\mathcal{M}_{\bm{X}}=\alpha\sqrt{1-\alpha^{2}}\left(\\!\\!\begin{array}[]{cccc}&&&\cos^{n}r\cos^{m}w\\\
&&0&\\\ &\mathinner{\mskip 1.0mu\raise 1.0pt\vbox{\kern 7.0pt\hbox{.}}\mskip
2.0mu\raise 4.0pt\hbox{.}\mskip 2.0mu\raise 7.0pt\hbox{.}\mskip 1.0mu}&&\\\
0&&&\\\ \end{array}\\!\\!\right).$ (38)

Finally, the sub-density operator $\rho_{\bm{B}}$ can be demonstrated as sub-
matrix $\mathcal{M}_{\bm{B}}$ that corresponds to elements

$\displaystyle|1^{1},0^{2}_{A},...,0^{n}_{A},0^{1}_{B},...,0^{m}_{B}\rangle\langle
1^{1},0^{2}_{A},...,0^{n}_{A},0^{1}_{B},...,0^{m}_{B}|:\rho_{2^{N-1}+1}$
$\displaystyle|1^{1},0^{2}_{A},...,0^{n}_{A},0^{1}_{B},...,0^{m-1}_{B},1^{m}_{B}\rangle\langle
1^{1},0^{2}_{A},...,0^{n}_{A},0^{1}_{B},...,0^{m-1}_{B},1^{m}_{B}|:\rho_{2^{N-1}+2},...,$
$\displaystyle|1^{1},...,1^{n}_{A},1^{1}_{B},...,1^{m-1}_{B},0^{m}_{B}\rangle\langle
1^{1},...,1^{n}_{A},1^{1}_{B},...,1^{m-1}_{B},0^{m}_{B}|:\rho_{2^{N}-1},$
$\displaystyle|1^{1}_{A},...,1^{n}_{A},1^{1}_{B},...1^{m}_{B}\rangle\langle
1^{1}_{A},...,1^{n}_{A},1^{1}_{B},...1^{m}_{B}|:\rho_{2^{N}},$

where

$\displaystyle\rho_{2^{N-1}+1}=\alpha^{2}\cos^{2(n-1)}r\sin^{2}r\cos^{2m}w,\rho_{2^{N-1}+2}=\alpha^{2}\cos^{2(n-1)}r\sin^{2}r\cos^{2(m-1)}w\sin^{2}w,$
$\displaystyle\rho_{2^{N}-1}=\alpha^{2}\sin^{2n}r\cos^{2}w\sin^{2(m-1)}w,\rho_{2^{N}}=\alpha^{2}\sin^{2n}r\sin^{2m}w+1-\alpha^{2}.$

Then, the sub-matrix $\mathcal{M}_{\bm{B}}$ can be expressed as

$\displaystyle\mathcal{M}_{\bm{B}}=\left(\\!\\!\begin{array}[]{ccccc}\rho_{2^{N-1}+1}&&&&\\\
&\rho_{2^{N-1}+2}&&&\\\ &&\ddots&\\\ &&&\rho_{2^{N}-1}&\\\ &&&&\rho_{2^{N}}\\\
\end{array}\\!\\!\right).$ (44)


# Two-Loop Electron Factor Contribution to Lamb Shift in Muonium and
Positronium

Michael I. Eides<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Kentucky, Lexington, KY 40506, USA  Valery A. Shelyuto
<EMAIL_ADDRESS>D. I. Mendeleyev Institute for Metrology, St.Petersburg
190005, Russia

###### Abstract

We calculate hard spin-independent contributions to energy levels in muonium
and positronium which are due to radiatively corrected electron factor
insertion in two-photon exchange diagrams. Calculation of these corrections is
motivated by the new round of precise measurements of spin-independent
transition frequencies in muonium and positronium.

For many years experimental and theoretical research on energy levels in
muonium and positronium concentrated on hyperfine structure, see reviews in
[1, 2, 3, 4, 5]. Now a new generation of experiments on measuring spin-
independent transitions ($1S-2S$, $2S-2P$, etc.) in muonium and positronium
(see, e.g., [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 5]) is either going
on or planned. Inspired by these new developments we recently started a
program of calculating hard three-loop spin-independent corrections to energy
levels in muonium and positronium [18, 19].

There are numerous gauge invariant sets of diagrams generating such
corrections. We have already calculated contributions of three gauge invariant
sets of diagrams in Fig. 1 [18, 19].

Figure 1:

Below we calculate contributions to the Lamb shift in muonium and positronium
generated by one more set of diagrams in Fig. 2 (plus diagrams with crossed
exchanged photons, which are not shown explicitly). In the case of muonium
nonrecoil contribution of these diagrams was calculated long time ago [20], so
here we calculate only the radiative-recoil contribution.

For any system of two electromagnetically interacting leptons with unequal (or
equal) masses the hard spin-independent energy shift to the bound state energy
level generated by the diagrams with two-photon exchanges is described by the
integral [21]

$\Delta E=-\frac{(Z\alpha)^{5}}{\pi
n^{3}}m_{r}^{3}\int{\frac{d^{4}k}{i\pi^{2}k^{4}}}\frac{1}{4}Tr\Bigl{[}(1+\gamma_{0})L_{\mu\nu}\Bigr{]}\frac{1}{4}Tr\Bigl{[}(1+\gamma_{0})H_{\mu\nu}\Bigr{]}\delta_{l0},$
(1)

where $m$ and $M$ are the masses of the leptons, $L_{\mu\nu}$ and $H_{\mu\nu}$
are the light and heavy fermion factors, respectively, $m_{r}=mM/(m+M)$ is the
reduced mass, $Z=1$ is the charge of the heavy fermion in terms of the
positron charge, $n$ and $l$ are the principal quantum number and the orbital
momentum, respectively. The expression in Eq. (1) is exact in the mass ratio,
and is valid also in the case of $m=M$ (positronium).

Figure 2:

The radiatively corrected electron factor is a sum of three terms

$L_{\mu\nu}=L_{\mu\nu}^{\Sigma}+2L_{\mu\nu}^{\Lambda}+L_{\mu\nu}^{\Xi},$ (2)

arising from the two-loop self-energy, vertex and spanning photon insertions
in electron line and corresponding to the diagrams in Fig. 3. Respectively,
the first trace in Eq. (1) can be written as

$\frac{1}{4}Tr\Bigl{[}(1+\gamma_{0})L_{\mu\nu}\Bigr{]}\equiv\frac{\alpha^{2}}{\pi^{2}m}{\cal
L}_{\mu\nu}\left(\frac{k}{m}\right)=\frac{\alpha^{2}}{\pi^{2}m}\left[{\cal
L}_{\mu\nu}^{\Sigma}\left(\frac{k}{m}\right)+2{\cal
L}_{\mu\nu}^{\Lambda}\left(\frac{k}{m}\right)+{\cal
L}_{\mu\nu}^{\Xi}\left(\frac{k}{m}\right)\right].$ (3)

Figure 3:

A photon line with the insertion of a one-loop polarization operator in Fig. 3
has a natural interpretation as a massive photon propagator with the mass
squared $\lambda^{2}=4m^{2}/(1-v^{2})$. The diagrams with the polarization
insertions are obtained from this massive propagator integrating over $v$ with
the weight $(\alpha/\pi)v^{2}(1-v^{2}/3)/(1-v^{2})$. All entries in the two-
loop fermion factor except the two-loop anomalous magnetic moment and two-loop
slope of the electric form factor decrease at least as $k^{2}$ at $k^{2}\to
0$. As a result, the slowly decreasing terms with the anomalous magnetic
moment and slope of the electric form factor produce infrared-divergent
contributions in the integral in Eq. (1) for the diagrams in Fig. 2. This
linear infrared divergence indicates existence of a contribution to the Lamb
shift of the previous order in $Z\alpha$ that is already well known. To get
rid of this spurious divergence, we subtract the terms with the two-loop
anomalous magnetic moment and slope of the electric form factor from the two-
loop electron factor.

The heavy line factor in Eq. (1) has the form

$H_{\mu\nu}=\gamma_{\mu}\frac{\hat{P}+\hat{k}+M}{k^{2}+2Mk_{0}+i0}\gamma_{\nu}+\gamma_{\nu}\frac{\hat{P}-\hat{k}+M}{k^{2}-2Mk_{0}+i0}\gamma_{\mu},$
(4)

where $P=(M,{\bf 0})$ is the momentum of the particle with mass $M$.

In the case of $m\ll M$ the heavy trace reduces to

$\begin{split}\frac{1}{4}Tr\Bigl{[}(1+\gamma_{0})H_{\mu\nu}\Bigr{]}&\to-\frac{1}{M}\biggl{[}k^{2}g_{\mu
0}g_{\nu 0}\wp\Bigl{(}\frac{1}{k_{0}^{2}}\Bigr{)}-\bigl{(}g_{\mu
0}k_{\nu}+g_{\nu 0}k_{\mu}\bigr{)}\frac{1}{k_{0}}+g_{\mu\nu}\biggr{]}\\\
&\equiv-\frac{1}{M}{\cal H}_{\mu\nu}(k)_{rec},\end{split}$ (5)

where ${\cal H}_{\mu\nu}(k)_{rec}$ is a dimensionless function, and
$\wp\Bigl{(}\frac{1}{k_{0}^{2}}\Bigr{)}$ is the principal value integral, see
[21, 22] for the definition and properties.

The linear in mass ratio radiative-recoil contribution is obtained from Eq.
(1) by the substitution in Eq. (5) [21]

$\Delta
E_{rec}=\frac{\alpha^{2}(Z\alpha)^{5}}{\pi^{3}n^{3}}\frac{m_{r}^{3}}{Mm}\int{\frac{d^{4}k}{i\pi^{2}k^{4}}}{\cal
L}_{\mu\nu}\left(\frac{k}{m}\right){\cal H}_{\mu\nu}(k)_{rec}.$ (6)

This expression will be used for calculation of the radiative-recoil
contribution of the diagrams in Fig. 2 in muonium.

We have shown in [19] that there exists a simple relationship between the
integrand for the linear in mass ratio radiative-recoil corrections in Eq. (6)
and the integrand for the total (recoil and nonrecoil) spin-independent
contribution in the case of equal masses $m=M$. Namely, it is sufficient to
let $m=M$ and make the substitution ${\cal H}_{\mu\nu}(k)_{rec}\to{\cal
H}_{\mu\nu}(k)_{tot}$, where

${\cal H}_{\mu\nu}(k)_{tot}={\cal
H}_{\mu\nu}(k)_{rec}\frac{k_{0}^{2}}{k_{0}^{2}-\frac{k^{4}}{4m^{2}}}=\frac{k^{2}g_{\mu
0}g_{\nu 0}-(g_{\mu 0}k_{\nu}+g_{\nu
0}k_{\mu}){k_{0}}+g_{\mu\nu}k_{0}^{2}}{k_{0}^{2}-\frac{k^{4}}{4m^{2}}}.$ (7)

Then the total contribution to the spin-independent energy shift of order
$\alpha^{7}m$ generated by the diagrams in Fig. 2 in the case of equal masses
is given by the integral

$\Delta
E_{tot}=2\frac{\alpha^{2}(Z\alpha)^{5}}{\pi^{3}n^{3}}\frac{m_{r}^{3}}{Mm}\int{\frac{d^{4}k}{i\pi^{2}k^{4}}}{\cal
L}_{\mu\nu}\left(\frac{k}{m}\right){\cal H}_{\mu\nu}(k)_{tot},$ (8)

where an extra factor $2$ reflects the possibility to make radiative
insertions in both fermion lines.

We calculated the energy shifts in the Feynman gauge for the radiative
photons. The linear infrared divergences, which, as was explained above,
indicate the presence of the contributions of the previous order in $Z\alpha$,
were omitted, and the spurious logarithmic infrared divergences cancelled in
the sum of diagrams in Fig. 2. Using Eq. (6) we obtain radiative-recoil
correction in muonium

$\Delta
E^{(Mu)}=\left(J^{(Mu)}_{\Sigma}+2J^{(Mu)}_{\Lambda}+J^{(Mu)}_{\Xi}\right)\frac{\alpha^{2}(Z\alpha)^{5}}{\pi^{3}n^{3}}\frac{m}{M}\left(\frac{m_{r}}{m}\right)^{3}m\delta_{l0}.$
(9)

Calculations are similar to the ones in [18, 19], and the infrared finite
contributions of the diagrams in Fig. 2 are as follows

$J^{(Mu)}_{\Sigma}=0.10602(3),\quad 2J^{(Mu)}_{\Lambda}=-0.07644(2),\quad
J^{(Mu)}_{\Xi}=0.07373(3).$ (10)

The total radiative-recoil contribution to the Lamb shift in muonium from the
diagrams in Fig. 2 is

$\Delta
E^{(Mu)}=0.10332(3)\frac{\alpha^{2}(Z\alpha)^{5}}{\pi^{3}n^{3}}\frac{m}{M}\left(\frac{m_{r}}{m}\right)^{3}m\delta_{l0}.$
(11)

To calculate the spin-independent contribution to the energy shift in
positronium we use the expression in Eq. (8)

$\Delta
E^{(Ps)}=\left(J^{(Ps)}_{\Sigma}+2J^{(Ps)}_{\Lambda}+J^{(Ps)}_{\Xi}\right)\frac{\alpha^{7}}{\pi^{3}n^{3}}\frac{m}{4}\delta_{l0},$
(12)

where the infrared finite contributions of separate diagrams in the Feynman
gauge are

$J^{(Ps)}_{\Sigma}=-0.11001(2),\quad 2J^{(Ps)}_{\Lambda}=-0.07969(2),\quad
J^{(Ps)}_{\Xi}=-0.32816(1).$ (13)

Finally, the contribution to the Lamb shift in positronium from the diagrams
in Fig. 2 is

$\Delta E^{(Ps)}=-0.12947(3)\frac{\alpha^{7}m}{\pi^{3}n^{3}}\delta_{l0}.$ (14)

Combining the results in Eq. (11) with our earlier results for muonium [18] we
obtain the total radiative-recoil contribution of the diagrams in Fig. 1 and
Fig. 2

$\Delta
E^{(Mu)}=-11.3275(2)\frac{\alpha^{2}(Z\alpha)^{5}}{\pi^{3}n^{3}}\frac{m}{M}\left(\frac{m_{r}}{m}\right)^{3}m\delta_{l0}.$
(15)

The contribution to the Lamb shift in positronium generated by the diagrams in
Fig. 1 [18, 19] and Fig. 2 is

$\Delta E^{(Ps)}=0.8057(2)\frac{\alpha^{7}m}{\pi^{3}n^{3}}\delta_{l0}.$ (16)

The contributions in Eq. (15) and Eq. (16) are too small to play a significant
role for the results of the ongoing experiments, they are at the level of a
few tenths of kHtz and a few kHz, respectively. However, we expect that these
corrections will become phenomenologically relevant in the future with further
improvements of the experimental accuracy.

There are other gauge-invariant sets of three-loop diagrams which arise as
radiative corrections to the two-photon exchange diagrams, see, e.g., [22].
Hard spin-dependent corrections generated by these diagrams are already
calculated, see, e.g., the review in [23] and references therein. Respective
spin-independent corrections remain at this time unknown, and we hope to
calculate them in the near future.

###### Acknowledgements.

Work of M. I. Eides was supported by the NSF grant PHY- 2011161.

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