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13 pages, 280 KiB  
Article
Intervention to Prevent Recurrent Intestinal Parasitic Infections in People Living with HIV in Selected Parts of Eastern Cape, South Africa
by Ifeoma Anozie, Mojisola Clara Hosu, Teke Apalata and Dominic T. Abaver
Trop. Med. Infect. Dis. 2024, 9(12), 289; https://doi.org/10.3390/tropicalmed9120289 - 27 Nov 2024
Viewed by 557
Abstract
Interactions between parasites and hosts are not fully understood, though the dynamic pattern of infection and reinfection in humans varies with different demographic variables and behavioral changes. A community-based non-equivalent control group post-test-only design, an aspect of quasi-experimental design (QED), was carried out [...] Read more.
Interactions between parasites and hosts are not fully understood, though the dynamic pattern of infection and reinfection in humans varies with different demographic variables and behavioral changes. A community-based non-equivalent control group post-test-only design, an aspect of quasi-experimental design (QED), was carried out between March 2019 and February 2020. For the extraction of data from respondents, structural questionnaires were filled. Their CD4 count and viral load from the database of the National Health Laboratory Services, Mthatha were recorded. The method applied for the identification of intestinal parasites was a direct examination of the stool and the use of concentration methods. The post-test analysis showed that the intervention sites that received THEdS (Treatment, Health education, and Sanitation) bundle had a cure proportion of 60% and a re-infection proportion of 40%. The post-test results on control sites (treatment-only group) showed that the cure proportion was 51.4% and the re-infection proportion was 48.6%. The viral load significantly reduced from 377 to 44 copies/mL with a significant increment in CD4 count from 244 to 573 (cells µL) and (p-value) = 0.002. The combination of THEdS is an effective measure to reduce infection and reinfection of intestinal parasites. The THEdS bundle is a sustainable control and prevention method for the control of helminthes and protozoan associated with unsanitary environment and poor personal hygiene among immune-compromised individuals like HIV/AIDS patients. Full article
19 pages, 3381 KiB  
Review
TIME REFRACTION and SPACETIME OPTICS
by José Tito Mendonça
Symmetry 2024, 16(11), 1548; https://doi.org/10.3390/sym16111548 - 19 Nov 2024
Viewed by 782
Abstract
A review of recent advances in spacetime optics is given, with special emphasis on time refraction. This is a basic optical process, occurring at a temporal discontinuity or temporal boundary, which is able to produce various different effects, such as frequency shifts, energy [...] Read more.
A review of recent advances in spacetime optics is given, with special emphasis on time refraction. This is a basic optical process, occurring at a temporal discontinuity or temporal boundary, which is able to produce various different effects, such as frequency shifts, energy amplification, time reflection, and photon emission. If, instead of a single discontinuity, we have two reverse temporal boundaries, we can form a temporal beam splitter, where temporal interferences can occur. It will also be shown that, in the presence of an axis of symmetry, such as a magnetic field, the temporal beam splitter can induce a rotation of the initial polarization state, similar to a Faraday rotation. Recent work on time crystals, superluminal fronts, and superfluid light will be reviewed. Time gates based on spacetime optical effects will be discussed. We also mention recent work on optical metamaterials. Finally, the quantum properties of time refraction, which imply the emission of photon from vacuum, are considered, while similar problems in high-energy QED associated with electron–positron pairs are briefly mentioned. Full article
(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers 2024)
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Figure 1

Figure 1
<p>Spacetime refraction: (<b>a</b>) boundary moving with velocity <span class="html-italic">v</span> along the <span class="html-italic">x</span>-axis, (<b>b</b>) boundary at rest, <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. We represent the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> plane, where two optical media with different refractive indices, <math display="inline"><semantics> <msub> <mi>n</mi> <mn>0</mn> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>n</mi> <mn>1</mn> </msub> </semantics></math>, are separated by a moving boundary (oblique line in the plane). The angles represent the photon velocity in the two media, <math display="inline"><semantics> <mrow> <mo form="prefix">tan</mo> <mrow> <mo>(</mo> <msub> <mi>α</mi> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>n</mi> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math>. For the particular case of (<b>b</b>), where the boundary is at rest, we get a simpler picture, where the boundary coincides with the <span class="html-italic">x</span>-axis. In this case, we include two initial beams moving with the same frequency but in opposite directions.</p>
Full article ">Figure 2
<p><math display="inline"><semantics> <msup> <mi>T</mi> <mn>2</mn> </msup> </semantics></math> (red curves) and <math display="inline"><semantics> <msup> <mi>R</mi> <mn>2</mn> </msup> </semantics></math> (blue curves): as a function of the ratio <math display="inline"><semantics> <mrow> <msubsup> <mi>E</mi> <mn>0</mn> <mo>′</mo> </msubsup> <mo>/</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.75</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1.25</mn> </mrow> </semantics></math> (dashed).</p>
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<p>Temporal beam splitter, made of two successive time refraction events: First, at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, the refractive index suddenly changes, from <math display="inline"><semantics> <msub> <mi>n</mi> <mn>0</mn> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>n</mi> <mn>1</mn> </msub> </semantics></math>, and then at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mi>τ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, the refractive index returns to its initial value <math display="inline"><semantics> <msub> <mi>n</mi> <mn>0</mn> </msub> </semantics></math>. Incident, reflected, and transmitted rays across the temporal slab of duration <math display="inline"><semantics> <mi>τ</mi> </semantics></math> are represented.</p>
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<p>Time crystal, made of <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>≫</mo> <mn>1</mn> </mrow> </semantics></math> successive temporal beam splitter events with the same duration <math display="inline"><semantics> <mi>τ</mi> </semantics></math>, and the same refractive index <math display="inline"><semantics> <msub> <mi>n</mi> <mn>1</mn> </msub> </semantics></math>, on a background of refractive index <math display="inline"><semantics> <msub> <mi>n</mi> <mn>0</mn> </msub> </semantics></math>, separated by a time interval <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>T</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>Superluminal front: plasma frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>p</mi> </msub> <mo>/</mo> <msub> <mi>ω</mi> <mrow> <mi>p</mi> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> in the time frame, moving with velocity <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>=</mo> <msup> <mi>c</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>u</mi> </mrow> </semantics></math> with respect to the laboratory frame.</p>
Full article ">Figure 6
<p>Temporal Rydberg–EIT gate: Radiation intensity transmitted through the gas, <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mrow> <mi>t</mi> <mi>r</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (in bold), frequency shift <math display="inline"><semantics> <mrow> <mi>α</mi> <mrow> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>ω</mi> <mo>/</mo> <msub> <mi>ω</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, and Stark-induced detuning, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (dashed curve). Normalized units are used.</p>
Full article ">Figure 7
<p>Time refraction in a static medium: diffraction ring at (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>&lt;</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, where the nonlinearity of the medium is described by <math display="inline"><semantics> <msubsup> <mi>χ</mi> <mn>1</mn> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </msubsup> </semantics></math>, and at (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>&gt;</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, where the nonlinearity susceptibility is <math display="inline"><semantics> <msubsup> <mi>χ</mi> <mn>1</mn> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </msubsup> </semantics></math>.</p>
Full article ">
50 pages, 751 KiB  
Article
Non-Equilibrium Quantum Brain Dynamics: Water Coupled with Phonons and Photons
by Akihiro Nishiyama, Shigenori Tanaka and Jack Adam Tuszynski
Entropy 2024, 26(11), 981; https://doi.org/10.3390/e26110981 - 15 Nov 2024
Viewed by 687
Abstract
We investigate Quantum Electrodynamics (QED) of water coupled with sound and light, namely Quantum Brain Dynamics (QBD) of water, phonons and photons. We provide phonon degrees of freedom as additional quanta in the framework of QBD in this paper. We begin with the [...] Read more.
We investigate Quantum Electrodynamics (QED) of water coupled with sound and light, namely Quantum Brain Dynamics (QBD) of water, phonons and photons. We provide phonon degrees of freedom as additional quanta in the framework of QBD in this paper. We begin with the Lagrangian density QED with non-relativistic charged bosons, photons and phonons, and derive time-evolution equations of coherent fields and Kadanoff–Baym (KB) equations for incoherent particles. We next show an acoustic super-radiance solution in our model. We also introduce a kinetic entropy current in KB equations in 1st order approximation in the gradient expansion and show the H-theorem for self-energy in Hartree–Fock approximation. We finally derive conserved number density of charged bosons and conserved energy density in spatially homogeneous system. Full article
(This article belongs to the Section Quantum Information)
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Figure 1

Figure 1
<p>2-Particle-Irreducible loop diagrams in Hartree–Fock approximation labeled by (<bold>a</bold>–<bold>h</bold>). The diagrams (<bold>a</bold>,<bold>b</bold>) represent local terms, while diagrams from (<bold>c</bold>) to (<bold>h</bold>) represent non-local terms. Solid lines, wavy lines and curly lines represent propagation of charged bosons, photons, and phonons, respectively. The dark circles in vertices involve covariant derivatives such as <inline-formula><mml:math id="mm859"><mml:semantics><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>i</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mi>e</mml:mi><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>. The light circles in external lines represent background coherent fields <inline-formula><mml:math id="mm860"><mml:semantics><mml:msub><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mrow><mml:mi>a</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm861"><mml:semantics><mml:msub><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mrow><mml:mi>o</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm862"><mml:semantics><mml:msub><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mrow><mml:mi>a</mml:mi><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm863"><mml:semantics><mml:msub><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mrow><mml:mi>o</mml:mi><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm864"><mml:semantics><mml:mover accent="true"><mml:mi>ψ</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm865"><mml:semantics><mml:msup><mml:mover accent="true"><mml:mi>ψ</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo>*</mml:mo></mml:msup></mml:semantics></mml:math></inline-formula>. The <inline-formula><mml:math id="mm866"><mml:semantics><mml:mrow><mml:mi>a</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm867"><mml:semantics><mml:mrow><mml:mi>o</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm868"><mml:semantics><mml:mrow><mml:mi>a</mml:mi><mml:mi>L</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm869"><mml:semantics><mml:mrow><mml:mi>o</mml:mi><mml:mi>L</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> represent acoustic transverse, optical transverse, acoustic longitudinal, and optical longitudinal phonons, respectively.</p>
Full article ">Figure 2
<p>Acoustic super-radiance emitted in a radial direction via a microtubule.</p>
Full article ">Figure 3
<p>Additional 2-Particle-Irreducible loop diagram labeled by (i) in Hartree–Fock approximation. The dark circles in vertices involve covariant derivatives. The light circles in external lines represent background coherent fields <inline-formula><mml:math id="mm870"><mml:semantics><mml:msub><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mrow><mml:mi>a</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm871"><mml:semantics><mml:msub><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mrow><mml:mi>o</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>.</p>
Full article ">
11 pages, 295 KiB  
Article
Hybrid Boson Sampling
by Vitaly Kocharovsky
Entropy 2024, 26(11), 926; https://doi.org/10.3390/e26110926 - 30 Oct 2024
Viewed by 519
Abstract
We propose boson sampling from a system of coupled photons and Bose–Einstein condensed atoms placed inside a multi-mode cavity as a simulation process testing the quantum advantage of quantum systems over classical computers. Consider a two-level atomic transition far-detuned from photon frequency. An [...] Read more.
We propose boson sampling from a system of coupled photons and Bose–Einstein condensed atoms placed inside a multi-mode cavity as a simulation process testing the quantum advantage of quantum systems over classical computers. Consider a two-level atomic transition far-detuned from photon frequency. An atom–photon scattering and interatomic collisions provide interactions that create quasiparticles and excite atoms and photons into squeezed entangled states, orthogonal to the atomic condensate and classical field driving the two-level transition, respectively. We find a joint probability distribution of atom and photon numbers within a quasi-equilibrium model via a hafnian of an extended covariance matrix. It shows a sampling statistics that is ♯P-hard for computing, even if only photon numbers are sampled. Merging cavity-QED and quantum-gas technologies into a hybrid boson sampling setup has the potential to overcome the limitations of separate, photon or atom, sampling schemes and reveal quantum advantage. Full article
(This article belongs to the Special Issue Quantum Computing in the NISQ Era)
10 pages, 280 KiB  
Review
Pseudo-Quantum Electrodynamics: 30 Years of Reduced QED
by Eduardo C. Marino, Leandro O. Nascimento, Van Sérgio Alves and Danilo T. Alves
Entropy 2024, 26(11), 925; https://doi.org/10.3390/e26110925 - 30 Oct 2024
Cited by 1 | Viewed by 667
Abstract
Charged quasiparticles, which are constrained to move on a plane, interact by means of electromagnetic (EM) fields which are not subject to this constraint, living, thus, in three-dimensional space. We have, consequently, a hybrid situation where the particles of a given system and [...] Read more.
Charged quasiparticles, which are constrained to move on a plane, interact by means of electromagnetic (EM) fields which are not subject to this constraint, living, thus, in three-dimensional space. We have, consequently, a hybrid situation where the particles of a given system and the EM fields (through which they interact) live in different dimensions. Pseudo-Quantum Electrodynamics (PQED) is a U(1) gauge field theory that, despite being strictly formulated in two-dimensional space, precisely describes the real EM interaction of charged particles confined to a plane. PQED is completely different from QED(2 + 1), namely, Quantum Electrodynamics of a planar gauge field. It produces, for instance, the correct 1/r Coulomb potential between static charges, whereas QED(2 + 1) produces lnr potential. In spite of possessing a nonlocal Lagrangian, it has been shown that PQED preserves both causality and unitarity, as well as the Huygens principle. PQED has been applied successfully to describe the EM interaction of numerous systems containing charged particles constrained to move on a plane. Among these are p-electrons in graphene, silicene, and transition-metal dichalcogenides; systems exhibiting the Valley Quantum Hall Effect; systems inside cavities; and bosonization in (2 + 1)D. Here, we present a review article on PQED (also known as Reduced Quantum Electrodynamics). Full article
(This article belongs to the Special Issue PQED: 30 Years of Reduced Quantum Electrodynamics)
24 pages, 1486 KiB  
Article
Finite Nuclear Size Effect on the Relativistic Hyperfine Splittings of 2s and 2p Excited States of Hydrogen-like Atoms
by Katharina Lorena Franzke and Uwe Gerstmann
Foundations 2024, 4(4), 513-536; https://doi.org/10.3390/foundations4040034 - 1 Oct 2024
Viewed by 882
Abstract
Hyperfine splittings play an important role in quantum information and spintronics applications. They allow for the readout of the spin qubits, while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact knowledge is thus of prior relevance. [...] Read more.
Hyperfine splittings play an important role in quantum information and spintronics applications. They allow for the readout of the spin qubits, while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact knowledge is thus of prior relevance. In this work, we analytically investigate the relativistic effects on the hyperfine splittings of hydrogen-like atoms, including finite-size effects of the nucleis’ structure. We start from exact solutions of Dirac’s equation using different nuclear models, where the nucleus is approximated by (i) a point charge (Coulomb potential), (ii) a homogeneously charged full sphere, and (iii) a homogeneously charged spherical shell. Equivalent modelling has been done for the distribution of the nuclear magnetic moment. For the 1s ground state and 2s excited state of the one-electron systems H1, H2, H3, and He+3, the calculated finite-size related hyperfine shifts are quite similar for the different structure models and in excellent agreement with those estimated by comparing QED and experiment. This holds also in a simplified approach where relativistic wave functions from a Coulomb potential combined with spherical-shell distributed nuclear magnetic moments promises an improved treatment without the need for an explicit solution of Dirac’s equation within the nuclear core. Larger differences between different nuclear structure models are found in the case of the anisotropic 2p3/2 orbitals of hydrogen, rendering these excited states as promising reference systems for exploring the proton structure. Full article
(This article belongs to the Section Physical Sciences)
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Figure 1

Figure 1
<p><b>Left:</b> Potential <math display="inline"><semantics> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mmultiscripts> <mi mathvariant="normal">H</mi> <none/> <none/> <mprescripts/> <none/> <mn>1</mn> </mmultiscripts> </semantics></math> generated by a point charge <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and by homogeneously charged full spheres and spherical shells, both with <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi mathvariant="normal">H</mi> </msub> <mo>=</mo> <mn>0.83</mn> </mrow> </semantics></math> fm (radius taken from [<a href="#B24-foundations-04-00034" class="html-bibr">24</a>]). All three potentials match outside the nucleus, i.e., for <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>≥</mo> <msub> <mi>R</mi> <mi mathvariant="normal">H</mi> </msub> </mrow> </semantics></math>. <b>Right:</b> Radial wave function <math display="inline"><semantics> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> for the <math display="inline"><semantics> <mrow> <mn>1</mn> <mi>s</mi> </mrow> </semantics></math> state of the three hydrogen isotopes for the three potentials. Note that the difference due to the different nuclear charge models is similar to that caused by the different isotopes (isotope shift).</p>
Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> (<b>top</b>) for hydrogen <math display="inline"><semantics> <mmultiscripts> <mi mathvariant="normal">H</mi> <none/> <none/> <mprescripts/> <none/> <mn>1</mn> </mmultiscripts> </semantics></math>, deuterium <math display="inline"><semantics> <mmultiscripts> <mi mathvariant="normal">H</mi> <none/> <none/> <mprescripts/> <none/> <mn>2</mn> </mmultiscripts> </semantics></math>, and tritium <math display="inline"><semantics> <mmultiscripts> <mi mathvariant="normal">H</mi> <none/> <none/> <mprescripts/> <none/> <mn>3</mn> </mmultiscripts> </semantics></math> for all three nuclei models. <math display="inline"><semantics> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> matches for all three potentials if <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>≥</mo> <mi>R</mi> </mrow> </semantics></math>. The inset shows the non-relativistic case (<math display="inline"><semantics> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) together with the relativistic form function of a pure Coulomb potential. The derivative <math display="inline"><semantics> <mrow> <msup> <mi>S</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <mi>s</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> of the relativistic form function is shown for hydrogen, deuterium, and tritium for all three nuclei models (<b>bottom</b>); <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>∂</mo> <mi>S</mi> </mrow> <mrow> <mo>∂</mo> <mi>r</mi> </mrow> </mfrac> </mstyle> </semantics></math> matches for all three potentials if <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>≥</mo> <mi>R</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>Electron probability <math display="inline"><semantics> <mrow> <msup> <mi>ψ</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>·</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> </semantics></math> (<span class="html-italic">r</span> in fm) of the excited <math display="inline"><semantics> <mrow> <mn>2</mn> <msub> <mi>p</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> state for the hydrogen isotopes <math display="inline"><semantics> <mmultiscripts> <mi mathvariant="normal">H</mi> <none/> <none/> <mprescripts/> <none/> <mn>1</mn> </mmultiscripts> </semantics></math>, <math display="inline"><semantics> <mmultiscripts> <mi mathvariant="normal">H</mi> <none/> <none/> <mprescripts/> <none/> <mn>2</mn> </mmultiscripts> </semantics></math>, <math display="inline"><semantics> <mmultiscripts> <mi mathvariant="normal">H</mi> <none/> <none/> <mprescripts/> <none/> <mn>3</mn> </mmultiscripts> </semantics></math>, and <math display="inline"><semantics> <mmultiscripts> <mi>He</mi> <none/> <mo>+</mo> <mprescripts/> <none/> <mn>3</mn> </mmultiscripts> </semantics></math> (curve for the latter multiplied by 500). For <math display="inline"><semantics> <mmultiscripts> <mi mathvariant="normal">H</mi> <none/> <none/> <mprescripts/> <none/> <mn>1</mn> </mmultiscripts> </semantics></math>, beside that for the full-sphere nucleus (solid lines) also the curve for the spherical-shell model (dashed line) is shown.</p>
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23 pages, 940 KiB  
Review
Overview of BK(∗)ℓℓ Theoretical Calculations and Uncertainties
by Farvah Mahmoudi and Yann Monceaux
Symmetry 2024, 16(8), 1006; https://doi.org/10.3390/sym16081006 - 7 Aug 2024
Cited by 6 | Viewed by 925
Abstract
The search for New Physics (NP) beyond the Standard Model (SM) has been a central focus of particle physics, including in the context of B-meson decays involving bs transitions. These transitions, mediated by flavour-changing neutral currents, are highly [...] Read more.
The search for New Physics (NP) beyond the Standard Model (SM) has been a central focus of particle physics, including in the context of B-meson decays involving bs transitions. These transitions, mediated by flavour-changing neutral currents, are highly sensitive to small NP effects due to their suppression in the SM. While direct searches at colliders have not yet led to NP discoveries, indirect probes through semi-leptonic decays have revealed anomalies in observables such as the branching fraction B(BKμμ) and the angular observable P5(BKμμ). In order to assess the observed tensions, it is essential to ensure an accurate SM prediction. In this review, we examine the theoretical basis of the BK() decays, addressing in particular key uncertainties arising from local and non-local form factors. We also discuss the impact of QED corrections to the Wilson coefficients, as well as the effect of CKM matrix elements on the predictions and the tension with the experimental measurements. We discuss the most recent results, highlighting ongoing efforts to refine predictions and to constrain potential signs of NP in these critical decay processes. Full article
(This article belongs to the Special Issue Symmetries and Anomalies in Flavour Physics)
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Figure 1

Figure 1
<p>Impact of local form factors on the prediction of (<b>a</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="script">B</mi> <mo>(</mo> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <mi>μ</mi> <mi>μ</mi> <mo>)</mo> </mrow> </semantics></math> with local form factors from [<a href="#B54-symmetry-16-01006" class="html-bibr">54</a>] (KR) and from [<a href="#B42-symmetry-16-01006" class="html-bibr">42</a>] (HPQCD); (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mi>P</mi> <mn>5</mn> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mn>0</mn> </msub> <mo>→</mo> <msup> <mi>K</mi> <mrow> <mo>∗</mo> <mn>0</mn> </mrow> </msup> <mi>μ</mi> <mi>μ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with local form factors from [<a href="#B58-symmetry-16-01006" class="html-bibr">58</a>] (GKvD) and from [<a href="#B53-symmetry-16-01006" class="html-bibr">53</a>] (BSZ). For both, non-local form factors from [<a href="#B20-symmetry-16-01006" class="html-bibr">20</a>], denoted as GRvDV, have been used.</p>
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<p>Impact of non-local form factors on the prediction of (<b>a</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="script">B</mi> <mo>(</mo> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <mi>μ</mi> <mi>μ</mi> <mo>)</mo> </mrow> </semantics></math> with local form factors from [<a href="#B42-symmetry-16-01006" class="html-bibr">42</a>] (HQPCD), and non-local form factors from [<a href="#B74-symmetry-16-01006" class="html-bibr">74</a>,<a href="#B75-symmetry-16-01006" class="html-bibr">75</a>] (QCDf) and from [<a href="#B20-symmetry-16-01006" class="html-bibr">20</a>] (GRvDV); (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mi>P</mi> <mn>5</mn> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mn>0</mn> </msub> <mo>→</mo> <msup> <mi>K</mi> <mrow> <mo>∗</mo> <mn>0</mn> </mrow> </msup> <mi>μ</mi> <mi>μ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with local form factors from [<a href="#B58-symmetry-16-01006" class="html-bibr">58</a>] (GKvD), and non-local form factors from [<a href="#B74-symmetry-16-01006" class="html-bibr">74</a>,<a href="#B75-symmetry-16-01006" class="html-bibr">75</a>] (QCDf) and from [<a href="#B20-symmetry-16-01006" class="html-bibr">20</a>] (GRvDV).</p>
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<p>Impact of QED corrections to Wilson coefficients on the prediction of (<b>a</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="script">B</mi> <mo>(</mo> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <mi>μ</mi> <mi>μ</mi> <mo>)</mo> </mrow> </semantics></math> with local form factors from [<a href="#B42-symmetry-16-01006" class="html-bibr">42</a>] (HPQCD) and non-local form factors from [<a href="#B20-symmetry-16-01006" class="html-bibr">20</a>] (GRvDV); (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mi>P</mi> <mn>5</mn> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mn>0</mn> </msub> <mo>→</mo> <msup> <mi>K</mi> <mrow> <mo>∗</mo> <mn>0</mn> </mrow> </msup> <mi>μ</mi> <mi>μ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with local form factors from [<a href="#B58-symmetry-16-01006" class="html-bibr">58</a>] (GKvD) and non-local form factors from [<a href="#B20-symmetry-16-01006" class="html-bibr">20</a>] (GRvDV).</p>
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<p>Impact of the CKM factor on the prediction of <math display="inline"><semantics> <mrow> <mi mathvariant="script">B</mi> <mo>(</mo> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <mi>μ</mi> <mi>μ</mi> <mo>)</mo> </mrow> </semantics></math> with local form factors from [<a href="#B42-symmetry-16-01006" class="html-bibr">42</a>] (HPQCD) and non-local form factors from [<a href="#B20-symmetry-16-01006" class="html-bibr">20</a>] (GRvDV).</p>
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16 pages, 7194 KiB  
Article
Structure-Based Design, Virtual Screening, and Discovery of Novel Patulin Derivatives as Biogenic Photosystem II Inhibiting Herbicides
by He Wang, Jing Zhang, Yu Ji, Yanjing Guo, Qing Liu, Yuan Chang, Sheng Qiang and Shiguo Chen
Plants 2024, 13(12), 1710; https://doi.org/10.3390/plants13121710 - 20 Jun 2024
Viewed by 1112
Abstract
Computer-aided design usually gives inspirations and has become a vital strategy to develop novel pesticides through reconstructing natural lead compounds. Patulin, an unsaturated heterocyclic lactone mycotoxin, is a new natural PSII inhibitor and shows significant herbicidal activity to various weeds. However, some evidence, [...] Read more.
Computer-aided design usually gives inspirations and has become a vital strategy to develop novel pesticides through reconstructing natural lead compounds. Patulin, an unsaturated heterocyclic lactone mycotoxin, is a new natural PSII inhibitor and shows significant herbicidal activity to various weeds. However, some evidence, especially the health concern, prevents it from developing as a bioherbicide. In this work, molecular docking and toxicity risk prediction are combined to construct interaction models between the ligand and acceptor, and design and screen novel derivatives. Based on the analysis of a constructed patulin–Arabidopsis D1 protein docking model, in total, 81 derivatives are designed and ranked according to quantitative estimates of drug-likeness (QED) values and free energies. Among the newly designed derivatives, forty-five derivatives with better affinities than patulin are screened to further evaluate their toxicology. Finally, it is indicated that four patulin derivatives, D3, D6, D34, and D67, with higher binding affinity but lower toxicity than patulin have a great potential to develop as new herbicides with improved potency. Full article
(This article belongs to the Special Issue Bioherbicide Development for Weed Control II)
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<p>Flow chart illustrating the structure-based ligand design and discovery of novel patulin derivatives with high herbicidal activity.</p>
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<p>Simulated modeling of patulin binding to the D1 protein of <span class="html-italic">Arabidopsis</span>. (<b>A</b>) The chemical structure of patulin. (<b>B</b>) Hydrogen bonding interactions of patulin binding to the D1 protein. (<b>C</b>) The stereo view of the patulin binding environment of the D1 protein, in which carbon, oxygen, nitrogen, and hydrogen atoms are displayed in gray, red, blue, and white, respectively. The green dashed lines represent the possible hydrogen bonds. (<b>D</b>) The surface representation of the Q<sub>B</sub> binding site with bound patulin.</p>
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<p>Binding interactions of patulin derivatives at the Q<sub>B</sub> binding site of D1 protein of <span class="html-italic">Arabidopsis</span>. An illustration of the binding mode of compounds D3 (<b>A</b>), D6 (<b>D</b>), D34 (<b>G</b>), and D67 (<b>J</b>) binding to the D1 protein, respectively. Key interaction types are represented in the color code. The stereo view of compound D3 (<b>B</b>), D6 (<b>E</b>), D34 (<b>H</b>), and D67 (<b>K</b>) binding environments at the Q<sub>B</sub> binding site. The surface representation of the Q<sub>B</sub> binding site with compounds D3 (<b>C</b>), D6 (<b>F</b>), D34 (<b>I</b>), and D67 (<b>L</b>), respectively.</p>
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14 pages, 1690 KiB  
Article
Present Status of Spectroscopy of the Hyperfine Structure and Repolarization of Muonic Helium Atoms at J-PARC
by Seiso Fukumura, Patrick Strasser, Mahiro Fushihara, Yu Goto, Takashi Ino, Ryoto Iwai, Sohtaro Kanda, Shiori Kawamura, Masaaki Kitaguchi, Shoichiro Nishimura, Takayuki Oku, Takuya Okudaira, Hirohiko M. Shimizu, Koichiro Shimomura, Hiroki Tada and Hiroyuki A. Torii
Physics 2024, 6(2), 877-890; https://doi.org/10.3390/physics6020054 - 12 Jun 2024
Viewed by 1064
Abstract
The mass mμ of the negative muon is one of the parameters of the elementary particle Standard Model and it allows us to verify the CPT (charge–parity–time) symmetry theorem by comparing mμ value with the mass mμ+ [...] Read more.
The mass mμ of the negative muon is one of the parameters of the elementary particle Standard Model and it allows us to verify the CPT (charge–parity–time) symmetry theorem by comparing mμ value with the mass mμ+ of the positive muon. However, the experimental determination precision of mμ is 3.1ppm, which is an order of magnitude lower than the determination precision of mμ+ at 120ppb. The authors aim to determine mμ and the magnetic moment μμ with a precision of O(10ppb) through spectroscopy of the hyperfine structure (HFS) of muonic helium-4 atom (4Heμe) under high magnetic fields. He4μe is an exotic atom where one of the two electrons of the He4 atom is replaced by a negative muon. To achieve the goal, it is necessary to determine the HFS of He4μe with a precision of O(1ppb). This paper describes the determination procedure of the HFS of He4μe in weak magnetic fields reported recently, and the work towards achieving the goal of higher precision measurement. Full article
(This article belongs to the Special Issue Precision Physics and Fundamental Physical Constants (FFK 2023))
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Figure 1
<p>The shape of asymmetry signal, <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>(</mo> <mo>Δ</mo> <mi>ω</mi> <mo>)</mo> </mrow> </semantics></math> (<a href="#FD7-physics-06-00054" class="html-disp-formula">7</a>), for different Rabi frequency <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>b</mi> <mo>|</mo> </mrow> </semantics></math> and [<math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>2000</mn> <mspace width="4pt"/> <mi>ns</mi> <mo>,</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mrow> <mn>60</mn> <mo>,</mo> <mn>000</mn> </mrow> <mspace width="4pt"/> <mi>ns</mi> </mrow> </semantics></math>] time interval. The value of the muon decay rate <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>≈</mo> <mn>455</mn> <mspace width="4pt"/> <mi>kHz</mi> </mrow> </semantics></math> was used.</p>
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<p>The shape of asymmetry signal, <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>(</mo> <mo>Δ</mo> <mi>ω</mi> <mo>)</mo> </mrow> </semantics></math> (<a href="#FD7-physics-06-00054" class="html-disp-formula">7</a>), for different [<math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> </semantics></math>] time intervals and the Rabi frequency <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>b</mi> <mo>|</mo> <mo>=</mo> <mn>600</mn> </mrow> </semantics></math> kHz. The value of the muon decay rate <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>≈</mo> <mn>455</mn> <mspace width="4pt"/> <mi>kHz</mi> </mrow> </semantics></math> was used.</p>
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<p>Schematic view of the experimental setup to measure the ground-state muonic helium atom HFS at zero magnetic field. The muon beam supplied by the J-PARC MUSE D-line is almost 100% polarized. A 1 mm thick copper beam stopper (not shown) is affixed on the Al absorber. The gas chamber can contain gases at pressures up to 10 atm. When exchanging target gases or changing pressures, the gas chamber is first evacuated. During this process, the pressure inside the small vacuum chamber is kept lower than that in the gas chamber to prevent deformation and damage to the CuBe windows. Reproduced from Ref. [<a href="#B3-physics-06-00054" class="html-bibr">3</a>], with permission from American Physical Society.</p>
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<p>The result of extrapolating the muonic helium atom HFS frequency to vacuum (solid circles). The solid line shows the linear fit result. The earlier results from Refs. [<a href="#B19-physics-06-00054" class="html-bibr">19</a>] (open circle) and [<a href="#B20-physics-06-00054" class="html-bibr">20</a>] (open squares) along with the linear extrapolation (dashed line) measured with He + Xe (1.5% doped) are shown for comparison (reproduced from Ref. [<a href="#B3-physics-06-00054" class="html-bibr">3</a>], with permission from American Physical Society).</p>
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<p>Measurements (black dots) and fit resonance curve (red line) at zero field, 10.4 atm, for the muonic helium atom (<b>a</b>) and muonium (<b>b</b>).The integral range is [1.6 µs, 60.0 µs], where the time origin is set to the second muon pulse arrival time. Note that the vertical scales of 10 times difference. The tables show the fitting procedure statistics.</p>
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<p>The distribution of decay positron detection position at the first layer of the detector when the microwave frequency is closest to the resonance frequency in muonium HFS measurements with (red line) and without (blue line) applying microwave being projected on the X (<b>left</b>) and Y (<b>right</b>) axes.</p>
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<p>Schematic view of muonic helium SEOP experiment. <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>He</mi> <none/> <none/> <mprescripts/> <none/> <mn>4</mn> </mmultiscripts> <msup> <mi>μ</mi> <mo>−</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> </mrow> </semantics></math> atoms are polarized by charge exchange or spin exchange between alkali metal atoms. The irradiation of the laser and the injection of muons are performed simultaneously. Ten forward/backward detectors are installed on the circumference centered on the beam axis, respectively. The shape of the glass cell was chosen to minimize the stress caused by the internal pressure. See text for further details.</p>
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<p>Simulation results with 5% of initial polarization assuming muonic helium and Rb atoms at 200 °C with no microwave resonance and no SEOP (black circles), with microvawe resonance and no SEOP (blue circles)), with no microwave resonance but with SEOP (pink circles), and with microwave resonance and SEOP (red circles).</p>
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28 pages, 3508 KiB  
Review
On Casimir and Helmholtz Fluctuation-Induced Forces in Micro- and Nano-Systems: Survey of Some Basic Results
by Daniel Dantchev
Entropy 2024, 26(6), 499; https://doi.org/10.3390/e26060499 - 7 Jun 2024
Cited by 6 | Viewed by 1449
Abstract
Fluctuations are omnipresent; they exist in any matter, due either to its quantum nature or to its nonzero temperature. In the current review, we briefly cover the quantum electrodynamic Casimir (QED) force as well as the critical Casimir (CC) and Helmholtz (HF) forces. [...] Read more.
Fluctuations are omnipresent; they exist in any matter, due either to its quantum nature or to its nonzero temperature. In the current review, we briefly cover the quantum electrodynamic Casimir (QED) force as well as the critical Casimir (CC) and Helmholtz (HF) forces. In the QED case, the medium is usually a vacuum and the massless excitations are photons, while in the CC and HF cases the medium is usually a critical or correlated fluid and the fluctuations of the order parameter are the cause of the force between the macroscopic or mesoscopic bodies immersed in it. We discuss the importance of the presented results for nanotechnology, especially for devising and assembling micro- or nano-scale systems. Several important problems for nanotechnology following from the currently available experimental findings are spelled out, and possible strategies for overcoming them are sketched. Regarding the example of HF, we explicitly demonstrate that when a given integral quantity characterizing the fluid is conserved, it has an essential influence on the behavior of the corresponding fluctuation-induced force. Full article
(This article belongs to the Collection Foundations of Statistical Mechanics)
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Figure 1
<p>The setup of the system considered by Casimir in his original article [<a href="#B1-entropy-26-00499" class="html-bibr">1</a>].</p>
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<p>The basic setup for discussing the thermodynamic Casimir effect [<a href="#B56-entropy-26-00499" class="html-bibr">56</a>].</p>
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<p>One-dimensional Ising model chain in a ring form. This is equivalent to a system with periodic boundary conditions. In the considered example, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, i.e., the number of “blue” atoms (molecules) is with 4 more than the number of “red” ones. It is also possible to consider that, say, the blue dots represent spins “up”, i.e., <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math>, while the red ones represents spins “down”, i.e., <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
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<p>One-dimensional Ising model chain in a ring form and one opposite (or defect) bond. This is equivalent to a system with antiperiodic boundary conditions. In the considered example, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, i.e., the number of “blue” atoms (molecules) is 4 more than the number of “red” ones. As in the periodic case, it is possible to consider the blue dots as depicting spins “up”, i.e., <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math>, and red ones as representing spins “down”, i.e., <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
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<p>Behavior of the function <math display="inline"><semantics> <mrow> <msubsup> <mi>X</mi> <mrow> <mi mathvariant="normal">H</mi> </mrow> <mrow> <mo>(</mo> <mi>per</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>K</mi> <mo>,</mo> <mi>m</mi> <mo>|</mo> <mi>N</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (see Equation (<a href="#FD36-entropy-26-00499" class="html-disp-formula">36</a>)) with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>100</mn> <mo>,</mo> <mn>200</mn> <mo>,</mo> <mn>300</mn> <mo>,</mo> <mn>400</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math>. It can be observed that the function is <span class="html-italic">positive</span> for large and sufficiently small values of <span class="html-italic">K</span>, while being <span class="html-italic">negative</span> for relatively moderate values of <span class="html-italic">K irrespective</span> of the value of <span class="html-italic">N</span>. The larger the value of <span class="html-italic">N</span>, the stronger the repulsion for a small enough <span class="html-italic">K</span>; in the latter regime, the force is strongly repulsive irrespective of the value of <span class="html-italic">N</span>.</p>
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<p>Behavior of the scaling function <math display="inline"><semantics> <mrow> <msubsup> <mi>X</mi> <mrow> <mi mathvariant="normal">H</mi> </mrow> <mrow> <mo>(</mo> <mi>per</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>t</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. Inspection of the results obtained numerically from Equation (29) with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>100</mn> <mo>,</mo> <mn>200</mn> <mo>,</mo> <mn>300</mn> <mo>,</mo> <mn>400</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math> along with those from Equation (<a href="#FD37-entropy-26-00499" class="html-disp-formula">37</a>) demonstrate perfect scaling and agreement. It can be observed that the function is <span class="html-italic">positive</span> for large values of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math>, <span class="html-italic">negative</span> for relatively moderate values of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math>, and again strongly repulsive for small values of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math>.</p>
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<p>Relief plot of the scaling function <math display="inline"><semantics> <mrow> <msubsup> <mi>X</mi> <mrow> <mi>Cas</mi> </mrow> <mrow> <mo>(</mo> <mi>per</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>t</mi> </msub> <mo>,</mo> <msub> <mi>x</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> for PBC (see Equation (<a href="#FD38-entropy-26-00499" class="html-disp-formula">38</a>)). The function is always <span class="html-italic">negative</span>, corresponding to an <span class="html-italic">attractive</span> force symmetric about <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>The figure shows the behavior of the function <math display="inline"><semantics> <mrow> <msubsup> <mi>X</mi> <mi>H</mi> <mrow> <mo>(</mo> <mi>per</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>t</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for PBC; see Equation (<a href="#FD37-entropy-26-00499" class="html-disp-formula">37</a>).</p>
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<p>Relief plot of the scaling function <math display="inline"><semantics> <mrow> <msubsup> <mi>X</mi> <mrow> <mi>Cas</mi> </mrow> <mrow> <mo>(</mo> <mi>anti</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>t</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math> for ABC (see Equation (<a href="#FD40-entropy-26-00499" class="html-disp-formula">40</a>)). Contrary to the periodic case, the force, is always <span class="html-italic">repulsive</span>.</p>
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<p>The figure shows the behavior of <math display="inline"><semantics> <mrow> <msubsup> <mi>X</mi> <mi>H</mi> <mrow> <mo>(</mo> <mi>anti</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>t</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for ABC; see Equation (<a href="#FD39-entropy-26-00499" class="html-disp-formula">39</a>).</p>
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<p>Behavior of the scaling function <math display="inline"><semantics> <mrow> <msubsup> <mi>X</mi> <mrow> <mi>Cas</mi> </mrow> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>t</mi> </msub> <mo>,</mo> <msub> <mi>x</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> of the Casimir force as a function of the scaling variables <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mi>h</mi> </msub> </semantics></math>. It can be observed that the function is <span class="html-italic">negative</span> for <span class="html-italic">all</span> values of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mi>h</mi> </msub> </semantics></math>.</p>
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<p>Behavior of the function <math display="inline"><semantics> <mrow> <msubsup> <mi>X</mi> <mrow> <mi mathvariant="normal">H</mi> </mrow> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>t</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. One observes that the force can be both attractive <span class="html-italic">and</span> repulsive depending on the values of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math> and <span class="html-italic">m</span>.</p>
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<p>Behavior of the scaling function <math display="inline"><semantics> <mrow> <msubsup> <mi>X</mi> <mrow> <mi>Cas</mi> </mrow> <mrow> <mo>(</mo> <mi>d</mi> <mi>b</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>t</mi> </msub> <mo>,</mo> <msub> <mi>x</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> of the Casimir force as a function of the scaling variables <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mi>h</mi> </msub> </semantics></math>. It can be observed that the function <span class="html-italic">changes sign</span> for a negative <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>a</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </semantics></math> depending on the values of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mi>h</mi> </msub> </semantics></math>.</p>
Full article ">Figure 14
<p>Behavior of the scaling function <math display="inline"><semantics> <mrow> <msubsup> <mi>X</mi> <mrow> <mi mathvariant="normal">H</mi> </mrow> <mrow> <mo>(</mo> <mi>d</mi> <mi>b</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>t</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the Helmholtz force. It can be observed that the force can be both attractive <span class="html-italic">and</span> repulsive depending on the values of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math> and <span class="html-italic">m</span>.</p>
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16 pages, 329 KiB  
Article
The Effective Potential of Scalar Pseudo-Quantum Electrodynamics in (2 + 1)D
by Leandro O. Nascimento, Carlos A. P. C. Junior and José R. Santos
Condens. Matter 2024, 9(2), 25; https://doi.org/10.3390/condmat9020025 - 30 May 2024
Viewed by 1301
Abstract
The description of the electron–electron interactions in two-dimensional materials has a dimensional mismatch, where electrons live in (2 + 1)D while photons propagate in (3 + 1)D. In order to define an action in (2 + 1)D, one may perform a dimensional reduction [...] Read more.
The description of the electron–electron interactions in two-dimensional materials has a dimensional mismatch, where electrons live in (2 + 1)D while photons propagate in (3 + 1)D. In order to define an action in (2 + 1)D, one may perform a dimensional reduction of quantum electrodynamics in (3 + 1)D (QED4) into pseudo-quantum electrodynamics (PQED). The main difference between this model and QED4 is the presence of a pseudo-differential operator in the Maxwell term. However, besides the Coulomb repulsion, electrons in a material are subjected to several microscopic interactions, which are inherent in a many-body system. These are expected to reduce the range of the Coulomb potential, leading to a short-range interaction. Here, we consider the coupling to a scalar field in PQED for explaining such a mechanism, which resembles the spontaneous symmetry breaking (SSB) in Abelian gauge theories. In order to do so, we consider two cases: (i) by coupling the quantum electrodynamics to a Higgs field in (3 + 1)D and, thereafter, performing the dimensional reduction; and (ii) by coupling a Higgs field to the gauge field in PQED and, subsequently, calculating its effective potential. In case (i), we obtain a model describing electrons interacting through the Yukawa potential and, in case (ii), we show that SSB does not occur at one-loop approximation. The relevance of the model for describing electronic interactions in two-dimensional materials is also addressed. Full article
(This article belongs to the Special Issue PQED: 30 Years of Reduced Quantum Electrodynamics)
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<p>The effective potential of the reduced-scalar model. We plot Equation (<a href="#FD21-condensedmatter-09-00025" class="html-disp-formula">21</a>) with <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Λ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>. Note that the local minimum in <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is the only acceptable ground state, whether we assume that <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> is always much less than <math display="inline"><semantics> <mi mathvariant="sans-serif">Λ</mi> </semantics></math>.</p>
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9 pages, 874 KiB  
Article
Penrose Scattering in Quantum Vacuum
by José Tito Mendonça
Photonics 2024, 11(5), 448; https://doi.org/10.3390/photonics11050448 - 10 May 2024
Viewed by 4942
Abstract
This paper considers the scattering of a probe laser pulse by an intense light spring in a QED vacuum. This new scattering configuration can be seen as the vacuum equivalent to the process originally associated with the scattering of light by a rotating [...] Read more.
This paper considers the scattering of a probe laser pulse by an intense light spring in a QED vacuum. This new scattering configuration can be seen as the vacuum equivalent to the process originally associated with the scattering of light by a rotating black hole, which is usually called Penrose superradiance. Here, the rotating object is an intense laser beam containing two different components of orbital angular momentum. Due to these two components having slightly different frequencies, the energy profile of the intense laser beam rotates with an angular velocity that depends on the frequency difference. The nonlinear properties of a quantum vacuum are described by a first-order Euler–Heisenberg Lagrangian. It is shown that in such a configuration, nonlinear photon–photon coupling leads to scattered radiation with frequency shift and angular dispersion. These two distinct properties, of frequency and propagation direction, could eventually be favorable for possible experimental observations. In principle, this new scattering configuration can also be reproduced in a nonlinear optical medium. Full article
(This article belongs to the Special Issue Photon-Photon Collision Using Extreme Lasers)
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<p>Geometry of Penrose scattering in vacuum: (<b>A</b>)—an intense light spring propagates in the <span class="html-italic">z</span> direction, with frequencies <math display="inline"><semantics> <msub> <mi>ω</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ω</mi> <mn>2</mn> </msub> </semantics></math>, its intensity rotates around the <span class="html-italic">z</span> axis; (<b>B</b>) a probe pulse with frequency <math display="inline"><semantics> <msub> <mi>ω</mi> <mi>i</mi> </msub> </semantics></math> propagates along the (negative) <span class="html-italic">x</span> direction and collides perpendicularly with the light spring; (<b>C</b>) scattered signals are emitted with frequencies <math display="inline"><semantics> <msub> <mi>ω</mi> <mo>±</mo> </msub> </semantics></math> and an angular spread dictated by the light spring structure.</p>
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<p>Representation of the radial integral <math display="inline"><semantics> <mrow> <mi mathvariant="script">I</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <msub> <mi>J</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in red, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in dashed red. We have used <math display="inline"><semantics> <mrow> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>14</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
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14 pages, 20281 KiB  
Article
Optimizing Josephson Junction Reproducibility in 30 kV E-Beam Lithography: An Analysis of Backscattered Electron Distribution
by Arthur M. Rebello, Lucas M. Ruela, Gustavo Moreto, Naiara Y. Klein, Eldues Martins, Ivan S. Oliveira, João P. Sinnecker and Francisco Rouxinol
Nanomaterials 2024, 14(9), 783; https://doi.org/10.3390/nano14090783 - 30 Apr 2024
Viewed by 1867
Abstract
This paper explores methods to enhance the reproducibility of Josephson junctions, which are crucial elements in superconducting quantum technologies, when employing the Dolan technique in 30 kV e-beam processes. The study explores the influence of dose distribution along the bridge area on reproducibility, [...] Read more.
This paper explores methods to enhance the reproducibility of Josephson junctions, which are crucial elements in superconducting quantum technologies, when employing the Dolan technique in 30 kV e-beam processes. The study explores the influence of dose distribution along the bridge area on reproducibility, addressing challenges related to fabrication sensitivity. Experimental methods include e-beam lithography, with electron trajectory simulations shedding light on the behavior of backscattered electrons. Wedescribe the fabrication of various Josephson junction geometries and analyze the correlation between the success rates of different lithography patterns and the simulated distribution of backscattered electrons. Our findings demonstrate a success rate of up to 96.3% for the double-resist 1-step low-energy e-beam lithography process. As a means of implementation strategy, we provide a geometric example that takes advantage of simulated stability regions to administer a controlled, uniform dose across the junction area, introducing novel features to overcome the difficulties associated with fabricating bridge-like structures. Full article
(This article belongs to the Section Nanofabrication and Nanomanufacturing)
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<p>Schematic representation of the pattern transferred onto a double-layer resist stack followed by an aluminum deposition process, delineating the steps involved in fabricating Josephson junctions, alongside electron trajectory simulation within the resist layers. (<b>a</b>) Dolan Josephson junction scheme showcasing the exposed area (in dark blue) and the central bridge region (in grey). (<b>b</b>) Bird’s-eye perspective of the anticipated Josephson junction bridge structure. (<b>c</b>) Initial <math display="inline"><semantics> <msup> <mn>30</mn> <mo>∘</mo> </msup> </semantics></math> angle deposition, (<b>d</b>) oxidation phase, (<b>e</b>) subsequent <math display="inline"><semantics> <mrow> <mo>−</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> angle deposition. (<b>f</b>) Representation of the Josephson junction post-lift-off process, with the green coating symbolizing the <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>l</mi> <msub> <mi>O</mi> <mi>x</mi> </msub> </mrow> </semantics></math> layer. (<b>g</b>) Visualization of electron dispersion trajectories in a 230 nm PMMA layer (<b>top</b>) and a 500 nm MMA co-polymer layer (<b>bottom</b>), both situated on a silicon substrate, under the influence of a 30 kV electron beam. The trajectories of the primary electrons from the incident beam are depicted in blue, whereas the backscattered electrons are illustrated in red.</p>
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<p>Statistical analysis of simulated backscattered electron trajectories, illustrating the correlation with angle and radius, accompanied by a schematic representation of the energy surface. (<b>a</b>) Simulated backscattered energy versus angle, with the green dashed line indicating the 2<math display="inline"><semantics> <mi>σ</mi> </semantics></math> confidence region. (<b>b</b>) Radius of the deposited energy surface for backscattered electrons as determined by simulation. First 300 nm section to be within the resist-material-selectivity threshold is from 60 to 360 nm, where the fitted backscattered energy will decay 75% from start to end, and this region is shaded in orange. The comparative 50% decay region is shaded in green. (<b>c</b>) The energy surface of the backscattered electrons (orange shade) surrounding the incident beam (red), integrated into a cross-sectional diagram of the bridge region. The pattern areas within 500 nm of the bridge section are highlighted in red, and areas within 1000 nm are shown in light red. The red arrow depicts the beam incident direction, the orange arrow shows backscattered electrons within the resist stack, and the green arrows indicate backscattered electrons permeating from within the substrate.</p>
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<p>(<b>a</b>) Designs employed to investigate the effects of backscattered electrons across different geometries, identified from left to right as thin Dolan, L, and horseshoe junction designs. (<b>b</b>) Resulting dose map from the integration of the Point Spread Function (PSF) over the thin geometry, with detailed analysis presented in panels (<b>e</b>,<b>f</b>). (<b>c</b>) Detailed view of the unexposed bridge region, with percentiles marking the total deposited dose per region. (<b>d</b>) Colored scanning electron microscope (SEM) image of an L-shaped Josephson junction, where the blue region indicates the first deposited aluminum layer and the <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>l</mi> <msub> <mi>O</mi> <mi>x</mi> </msub> </mrow> </semantics></math> tunneling barrier is highlighted in the center. (<b>e</b>) Total dose distribution profiles along the vertical trace, including some of the exposed region (200 nm on each side), (<b>f</b>) only 300 nm unexposed section, for horseshoe, L, and Thin Dolan geometry—percentiles here denote the range of maximum dose variation; the percentages on the legend in (<b>e</b>,<b>f</b>) are the ratio of energy deposited in the directly exposed areas to the indirectly exposed gap. (<b>g</b>) Angled colored SEM images showcasing the resist stack; on the left, the Thin Dolan pattern is inscribed without bridge formation, whereas on the right, the horseshoe pattern is exposed, clearly displaying the bridge structure. Green indicates Si substrate, blue PMMA resist surface, and orange for resist side walls seen at an angle.</p>
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<p>Analysis of dose factors and the ratio of the backscattered to total incident dose for various geometries investigated in this study. (<b>a</b>) Distribution of the backscattered electrons along a vertical trace. (<b>b</b>) Distribution along a horizontal trace. (<b>c</b>) Ratio of backscattered to incident dose over the vertical trace. (<b>d</b>) Dose variation observed in a thin Dolan geometry.</p>
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<p>Analysis of dose factors and the ratio of the backscattered to the total incident dose for the proposed geometry designed to utilize backscattered electrons for the undercut definition. (<b>a</b>) The logical basis to create a new geometry, specifically conceived to tailor the distribution of backscattered electrons, thereby minimizing variance across the junction area. The blue region is intended to receive the minimal necessary dose to develop the top resist layer, with backscattered electrons being generated within the green circle region by a higher dose factor. The larger circles represent a simplified model for the overlap of the backscattered regions, assuming point sources, with different colors indicating the degree of overlap. (<b>b</b>) X junction ggeometry designed with features to retain geometric resolution while achieving the (4:1) ratio for the (<b>c</b>) total deposited dose over the vertical trace. (<b>d</b>) Horizontal profile of the total deposited dose. (<b>e</b>) Ratio of backscattered to incident dose for the X junction.</p>
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<p>(<b>a</b>) Three-dimensional copper cavity. (<b>b</b>) Quantum Rabi map that provides a profile of the dynamic evolution of qubit states in a time-dependent landscape.</p>
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33 pages, 1038 KiB  
Article
QED Meson Description of the Anomalous Particles at ∼17 and ∼38 MeV
by Cheuk-Yin Wong
Universe 2024, 10(4), 173; https://doi.org/10.3390/universe10040173 - 7 Apr 2024
Cited by 1 | Viewed by 1599
Abstract
The Schwinger confinement mechanism stipulates that a massless fermion and a massless antifermion are confined as a massive boson when they interact in the Abelian QED interaction in (1+1)D.If we approximate light quarks as massless and apply the Schwinger confinement mechanism to quarks, [...] Read more.
The Schwinger confinement mechanism stipulates that a massless fermion and a massless antifermion are confined as a massive boson when they interact in the Abelian QED interaction in (1+1)D.If we approximate light quarks as massless and apply the Schwinger confinement mechanism to quarks, we can infer that a light quark and a light antiquark interacting in the Abelian QED interaction are confined as a QED meson in (1+1)D. Similarly, a light quark and a light antiquark interacting in the QCD interaction in the quasi-Abelian approximation will be confined as a QCD meson in (1+1)D. The QED and QCD mesons in (1+1)D can represent physical mesons in (3+1)D when the flux tube radius is properly taken into account. Such a theory leads to a reasonable description of the masses of π0,η, and η, and its extrapolation to the unknown QED sector yields an isoscalar QED meson at about 17 MeV and an isovector QED meson at about 38 MeV. The observations of the anomalous soft photons, the hypothetical X17 particle, and the hypothetical E38 particle bear promising evidence for the possible existence of the QED mesons. Pending further confirmation, they hold important implications on the properties on the quarks and their interactions. Full article
(This article belongs to the Special Issue Multiparticle Dynamics)
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Figure 1
<p>(<b>a</b>) <math display="inline"><semantics> <mrow> <mi>q</mi> <mover accent="true"> <mi>q</mi> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> production by the fusion of two virtual gluons in the de-excitation of a highly excited <math display="inline"><semantics> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mn>4</mn> </msup> </semantics></math>He<math display="inline"><semantics> <mrow> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>ABCD</mi> <mo>)</mo> </mrow> <msup> <mo>→</mo> <mn>4</mn> </msup> </mrow> </semantics></math>He (ground state) (A′B′CD) + <span class="html-italic">q</span><math display="inline"><semantics> <mover accent="true"> <mi>q</mi> <mo>¯</mo> </mover> </semantics></math>(X) with the fusion of two virtual gluons between B and A. (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>q</mi> <mover accent="true"> <mi>q</mi> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> production in hadron–hadron or a nucleus–nucleus collision by A + B → A′ + B′ + <math display="inline"><semantics> <mrow> <msup> <mrow> <mo>(</mo> <mi>q</mi> <mover accent="true"> <mi>q</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mo>→</mo> </mrow> </semantics></math> A′ + B′ +<math display="inline"><semantics> <mrow> <mstyle displaystyle="true"> <munder> <mo>∑</mo> <mi>i</mi> </munder> </mstyle> <msub> <mi>h</mi> <mi>i</mi> </msub> <mo>+</mo> <mi>q</mi> <mover accent="true"> <mi>q</mi> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
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<p>(<b>a</b>) A QED meson <span class="html-italic">X</span> can decay into two real photons <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>→</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mrow> </semantics></math>. (<b>b</b>) It can decay into two virtual photons, each of which subsequently decays into a <math display="inline"><semantics> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> <mo>)</mo> </mrow> </semantics></math> pair, <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>→</mo> <msubsup> <mi>γ</mi> <mn>1</mn> <mo>*</mo> </msubsup> <mo>+</mo> <msubsup> <mi>γ</mi> <mn>2</mn> <mo>*</mo> </msubsup> <mo>→</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>, and (<b>c</b>) it can decay into a single <math display="inline"><semantics> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> <mo>)</mo> </mrow> </semantics></math> pair, <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>→</mo> <msubsup> <mi>γ</mi> <mn>1</mn> <mo>*</mo> </msubsup> <mo>+</mo> <msubsup> <mi>γ</mi> <mn>2</mn> <mo>*</mo> </msubsup> <mo>→</mo> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> </mrow> </semantics></math>.</p>
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<p>The invariant mass distribution of the emitted <math display="inline"><semantics> <msup> <mi>e</mi> <mo>+</mo> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mi>e</mi> <mo>−</mo> </msup> </semantics></math> in the de-excitation of the compound nucleus <math display="inline"><semantics> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mn>4</mn> </msup> </semantics></math>He<math display="inline"><semantics> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mo>*</mo> </msup> </semantics></math> state at 20.49 MeV in the <math display="inline"><semantics> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mn>3</mn> </msup> </semantics></math>H(<math display="inline"><semantics> <mrow> <mi>p</mi> <mo>,</mo> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> <mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mn>4</mn> </msup> </mrow> </semantics></math>He<math display="inline"><semantics> <msub> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mi>gs</mi> </msub> </semantics></math> reaction at <math display="inline"><semantics> <mrow> <msubsup> <mi>E</mi> <mi>p</mi> <mi>lab</mi> </msubsup> <mo>=</mo> </mrow> </semantics></math> 0.9 MeV [<a href="#B38-universe-10-00173" class="html-bibr">38</a>]. Red data points are the data in the signal region <math display="inline"><semantics> <mrow> <mn>19.5</mn> <mo>&lt;</mo> <msub> <mi>E</mi> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> </mrow> </msub> <mrow> <mo>(</mo> <mi>sum</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> </mrow> </semantics></math> 22.0 MeV, and black points are data in the background region <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>&lt;</mo> <msub> <mi>E</mi> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> </mrow> </msub> <mrow> <mo>(</mo> <mi>sum</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> </mrow> </semantics></math> 19.0 MeV. The solid (green) curve is the fit to the invariant mass data points.</p>
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<p>Comparison of the ATOMKI experimental data [<a href="#B37-universe-10-00173" class="html-bibr">37</a>,<a href="#B39-universe-10-00173" class="html-bibr">39</a>,<a href="#B41-universe-10-00173" class="html-bibr">41</a>,<a href="#B44-universe-10-00173" class="html-bibr">44</a>] with the ATOMKI X17 emission model predictions of the minimum opening angle <math display="inline"><semantics> <msub> <mi>θ</mi> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> </mrow> </msub> </semantics></math>(min) between <math display="inline"><semantics> <msup> <mi>e</mi> <mo>+</mo> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mi>e</mi> <mo>−</mo> </msup> </semantics></math> as a function of the X17 kinetic energy <span class="html-italic">K</span> in the CM frame, for different collision energies, targets, and final states. The X17 emission model envisages the fusion of the incident proton <span class="html-italic">p</span> with the target nucleus <span class="html-italic">A</span>, forming a compound nucleus <math display="inline"><semantics> <msup> <mi>C</mi> <mo>*</mo> </msup> </semantics></math>, which subsequently de-excites to the final state <math display="inline"><semantics> <msub> <mi>C</mi> <mi mathvariant="normal">f</mi> </msub> </semantics></math> with the simultaneous emission of the X17 particle. The subsequent decay of the X17 particle into <math display="inline"><semantics> <msup> <mi>e</mi> <mo>+</mo> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mi>e</mi> <mo>−</mo> </msup> </semantics></math> then gives the angle <math display="inline"><semantics> <msub> <mi>θ</mi> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> </mrow> </msub> </semantics></math> between <math display="inline"><semantics> <msup> <mi>e</mi> <mo>+</mo> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mi>e</mi> <mo>−</mo> </msup> </semantics></math>. The solid curve is the theoretical prediction of <math display="inline"><semantics> <msub> <mi>θ</mi> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> </mrow> </msub> </semantics></math>(min) as a function of the X17 kinetic energy <span class="html-italic">K</span>, and the ATOMKI data points are from [<a href="#B37-universe-10-00173" class="html-bibr">37</a>,<a href="#B39-universe-10-00173" class="html-bibr">39</a>,<a href="#B41-universe-10-00173" class="html-bibr">41</a>,<a href="#B44-universe-10-00173" class="html-bibr">44</a>] as summarized in <a href="#universe-10-00173-t002" class="html-table">Table 2</a>.</p>
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<p>The diphoton invariant mass spectra from light ion collisions with C and Cu targets at a few GeV per nucleon at the JINR Nuclotron, Dubna [<a href="#B72-universe-10-00173" class="html-bibr">72</a>]. The solid curve in the upper panel shows the invariant mass distribution obtained by combining two photons from the same event, and green shaded region the invariant mass distribution by combining two photons from mixed events. The signal of correlated photons subtracting the mixed event background gives the signal represented by the blue region in the lower panel, where the resonance-like structures at ∼17 and ∼38 MeV show up.</p>
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<p>(<b>a</b>) The spectrum of the diphoton energy sum, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>N</mi> <mo>/</mo> <mi>d</mi> <msub> <mi>E</mi> <mrow> <mi>γ</mi> <mi>γ</mi> </mrow> </msub> </mrow> </semantics></math>, in the <math display="inline"><semantics> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mn>3</mn> </msup> </semantics></math>H<math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>γ</mi> <mi>γ</mi> <mo>)</mo> </mrow> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mn>4</mn> </msup> </mrow> </semantics></math>He<math display="inline"><semantics> <msub> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mi>gs</mi> </msub> </semantics></math> reaction with a proton beam energy at 1 MeV [<a href="#B47-universe-10-00173" class="html-bibr">47</a>], (<b>b</b>) the spectrum of the diphoton energy sum, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>N</mi> <mo>/</mo> <mi>d</mi> <msub> <mi>E</mi> <mrow> <mi>γ</mi> <mi>γ</mi> </mrow> </msub> </mrow> </semantics></math>, in the <math display="inline"><semantics> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mn>3</mn> </msup> </semantics></math>He<math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>γ</mi> <mi>γ</mi> <mo>)</mo> </mrow> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mn>4</mn> </msup> </mrow> </semantics></math>He<math display="inline"><semantics> <msub> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mi>gs</mi> </msub> </semantics></math> reactions using a cold neutron beam line [<a href="#B47-universe-10-00173" class="html-bibr">47</a>], and (<b>c</b>) the spectrum of the dilepton energy sum, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>N</mi> <mo>/</mo> <mi>d</mi> <msub> <mi>E</mi> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> </mrow> </msub> </mrow> </semantics></math>, in the <math display="inline"><semantics> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mn>3</mn> </msup> </semantics></math>H<math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> <mo>)</mo> </mrow> <msup> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mn>4</mn> </msup> </mrow> </semantics></math>He<math display="inline"><semantics> <msub> <mrow> <mo> </mo> <mspace width="-0.16em"/> </mrow> <mi>gs</mi> </msub> </semantics></math> reaction with a proton beam energy at 0.9 MeV, where the decay of X17 to <math display="inline"><semantics> <msup> <mi>e</mi> <mo>+</mo> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mi>e</mi> <mo>−</mo> </msup> </semantics></math> has been observed [<a href="#B39-universe-10-00173" class="html-bibr">39</a>].</p>
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23 pages, 4590 KiB  
Review
What Is the “Hydrogen Bond”? A QFT-QED Perspective
by Paolo Renati and Pierre Madl
Int. J. Mol. Sci. 2024, 25(7), 3846; https://doi.org/10.3390/ijms25073846 - 29 Mar 2024
Cited by 3 | Viewed by 1427
Abstract
In this paper we would like to highlight the problems of conceiving the “Hydrogen Bond” (HB) as a real short-range, directional, electrostatic, attractive interaction and to reframe its nature through the non-approximated view of condensed matter offered by a Quantum Electro-Dynamic (QED) perspective. [...] Read more.
In this paper we would like to highlight the problems of conceiving the “Hydrogen Bond” (HB) as a real short-range, directional, electrostatic, attractive interaction and to reframe its nature through the non-approximated view of condensed matter offered by a Quantum Electro-Dynamic (QED) perspective. We focus our attention on water, as the paramount case to show the effectiveness of this 40-year-old theoretical background, which represents water as a two-fluid system (where one of the two phases is coherent). The HB turns out to be the result of the electromagnetic field gradient in the coherent phase of water, whose vacuum level is lower than in the non-coherent (gas-like) fraction. In this way, the HB can be properly considered, i.e., no longer as a “dipolar force” between molecules, but as the phenomenological effect of their collective thermodynamic tendency to occupy a lower ground state, compatible with temperature and pressure. This perspective allows to explain many “anomalous” behaviours of water and to understand why the calculated energy associated with the HB should change when considering two molecules (water-dimer), or the liquid state, or the different types of ice. The appearance of a condensed, liquid, phase at room temperature is indeed the consequence of the boson condensation as described in the context of spontaneous symmetry breaking (SSB). For a more realistic and authentic description of water, condensed matter and living systems, the transition from a still semi-classical Quantum Mechanical (QM) view in the first quantization to a Quantum Field Theory (QFT) view embedded in the second quantization is advocated. Full article
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Molecular orbitals of the water molecule in LCAO theory of the isolated molecule. Note: the inner 1s orbital is not shown; σ denotes the bonding configuration; σ* denotes the anti-bonding configuration, which leads to molecular instability and thus splitting of the constituting atoms. The colours represent the positive (green/yellow) or negative (blue/red) value of the orbital wave function participating to the exchange integral describing the bonding character (positive product) or antibonding character (negative product) of the interaction (product) with other orbitals associated with the estimated locations of the electrons around the nuclei (Composite representation based on [<a href="#B24-ijms-25-03846" class="html-bibr">24</a>,<a href="#B25-ijms-25-03846" class="html-bibr">25</a>]).</p>
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<p>(<b>Left</b>) Panels: Gas phase Photo-Electron Spectroscopy (PES) spectrum of water measured at a photon energy of 100 eV (top left) and PES spectra of ice at photon energies of 100 eV and 530 eV (bottom left). X-ray Absorption Spectroscopy (XAS) spectra of gas phase water and ice together with a Density Functional Theory (DFT) calculation of hexagonal ice described by a 44-molecule cluster [<a href="#B26-ijms-25-03846" class="html-bibr">26</a>]. (<b>Right</b>) Sketch: Scheme showing that the 3a<sub>1</sub>-(HOMO-1) molecular orbital lies in the molecular plane, while the 1b<sub>1</sub>-(HOMO) orbital lies in a plane perpendicular to the σ-bonds (O–H). Yellow-blue pairing denotes anti-bonding, yellow-green pairing denotes bonding orbitals. The overlap is expressed by the exchange-integral S. Partial HB’s covalence is thus only possible if the fully occupied in-plane 3a<sub>1</sub>-(HOMO-1) level overlaps with an empty 4a<sub>1</sub>-(LUMO) of a neighbouring molecule. The overlap with the other out-of-plane 1b<sub>1</sub>-(HOMO) is zero. Strong HBs are then expected to interact with the 3a<sub>1</sub>-level while van der Waals interactions are expected for the 1b<sub>1</sub>-level. However, the changes in the 3a<sub>1</sub> orbital revealed by XES [<a href="#B26-ijms-25-03846" class="html-bibr">26</a>] are not experimental evidence for electron sharing (covalence) in HBs [<a href="#B29-ijms-25-03846" class="html-bibr">29</a>] because in (HOMO-LUMO) frontier-orbital theory the assumed covalence would primarily affect the HOMO outmost 1b<sub>1</sub> orbital and definitively not the 3a<sub>1</sub>-(HOMO-1). Covalence in a QM sense, i.e., HOMO/LUMO interaction, is thus not supported by experimental data. In other words, a valid CP or QM picture of HB seems impossible.</p>
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<p>The excitation spectrum [<a href="#B32-ijms-25-03846" class="html-bibr">32</a>,<a href="#B33-ijms-25-03846" class="html-bibr">33</a>] of water vapour (isolated molecules) in the visible and near UV range (up to 20 eV); the red line indicates (by considering the final density of the liquid at room P and T) the predicted excited level (5d orbital) that satisfies several favourable conditions at once, such as not too high critical density, sufficient oscillator strength, relevant coupling constants <span class="html-italic">g<sub>c</sub></span>, <span class="html-italic">μ<sub>r</sub></span> and energy gap.</p>
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<p>A scheme of the energetics of the coherent state of water molecules showing that such a state is the result of a continuous <span class="html-italic">collective</span> oscillation of the molecules between two states, driven by the self-trapped em-field whose phase (and renormalized frequency) is locked to the phase of the matter-field. Considering the coherent oscillation, the water molecules adopt two limit-shapes (excited and relaxed) with different time-weights during the oscillation cycle. The value of the energy gap, (Δ<span class="html-italic">g</span> ≈ 0.16 eV) in this scheme refers to the latest calculation done for liquid water, neglecting the temperature contribution [<a href="#B39-ijms-25-03846" class="html-bibr">39</a>]. However, as will be discussed in the following, the energy stability is not independent of the radial position within the CD (see Equation (13a,b) in the text), thus the energetic profile of the coherent ground state within CDs depends on the aggregation state and the thermodynamic conditions. The thermodynamic parameters (like pressure and temperature) can influence the establishment of other types of coherence [<a href="#B39-ijms-25-03846" class="html-bibr">39</a>], favouring the choice of other excited levels among those available in the electron spectrum of water (to which other field amplitude, oscillator strength, energy gap, coupling constant, critical density, renormalization frequency are associated). This is a key aspect to understand why the classical concept of “HB” depends on the aggregation state (water dimer, liquid, ice, supercooled clusters, etc.) [<a href="#B15-ijms-25-03846" class="html-bibr">15</a>]. A mixing angle α, which gives sin<sup>2</sup>(α) = 0.1 indicates that electrons in water molecules spend 10% of their time in the excited level (Eq, the 5d oxygen orbital), so that coherent water molecules are larger than incoherent water molecules. Such a fact can explain (i) the flickering landscape of intermolecular interactions (including the so-called “HBs”), as well as (ii) the evidence of tetrahedral structures in some regions of the liquid (or as in hexagonal ice and in confined water). Indeed, two of the five d-orbitals (z<sup>2</sup>, x<sup>2</sup> − y<sup>2</sup>) transform into the totally symmetric a1-representation of the C<sub>2v</sub> group and can mix themselves with the two other molecular orbitals (2a<sub>1</sub>, 3a<sub>1</sub>) giving rise to a set of four a<sub>1</sub>-type levels arranged in a more or less tetrahedral configuration to minimize electronic repulsions [<a href="#B40-ijms-25-03846" class="html-bibr">40</a>].</p>
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<p>On the left, an artist’s sketch to give an idea of the biphasic picture of liquid water: at room conditions the coherent fraction, <span class="html-italic">F<sub>coh</sub></span>(<span class="html-italic">T</span>), comprises about 40% of the total molecules and its density (0.92 g/cm<sup>3</sup>) is independent of temperature, while the density of the incoherent interstitial fraction depends on T. This requires, as shown in the right panel, modelling water density as a function of T, <span class="html-italic">ρ</span>(<span class="html-italic">T</span>). This can be done via the sum of two contributions (one for each fluid, coherent and bulk), as has been successfully demonstrated also for predicting trends of other properties in water systems (as isobar specific heat [<a href="#B38-ijms-25-03846" class="html-bibr">38</a>,<a href="#B54-ijms-25-03846" class="html-bibr">54</a>] viscosity [<a href="#B60-ijms-25-03846" class="html-bibr">60</a>] and electric susceptivity [<a href="#B61-ijms-25-03846" class="html-bibr">61</a>]).</p>
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<p>Artistic sketch to give an intuitive idea of the two different pictures emerging from corpuscular QM (<b>left</b>) and QFT-QED (<b>right</b>) views: within the former approach—where electrodynamic interactions are only perturbative—the cohesion and condensed state of liquid water is supposed to rest on a flickering network of local directional forces. This picture suffers from the inability to physically justify how water molecules could express a topology of their electron cloud that is protruding enough to yield directional electrostatic oriented pair-potentials. It also lacks an explanation of the physical reason why this flickering network in such a “mixture model” [<a href="#B80-ijms-25-03846" class="html-bibr">80</a>] should be divided into two populations (as required to fit the now common experimental evidence that liquid water is a two-phase system) [<a href="#B26-ijms-25-03846" class="html-bibr">26</a>]. On the right pane, an “instantaneous frame” representing liquid water at ordinary temperatures, where CDs (in blue), appearing and disappearing every few hundred of femtoseconds, are immersed in a stochastic, vapor-like, incoherent (denser) fraction, that becomes more and more abundant with increasing temperature. In the latter case, two types of dynamics co-exist, time-relaxations, kinetics, orderings, and geometries [<a href="#B40-ijms-25-03846" class="html-bibr">40</a>].</p>
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<p>On the left panel, an intuitive sketch to describe the CD as a region of space where, the confined decaying em-field (top scheme) couples with matter quanta, generating a new, energetically lower, vacuum level (where phase correlations operate thanks to the condensation of bosonic quasi-particles). In SSB, the massless part of these quasi-particles is called phasons [<a href="#B50-ijms-25-03846" class="html-bibr">50</a>] and to their space range a lower vacuum level is associated, constituting a potential well with respect to non-coherent region. The passage from a vacuum level to the other traces an energetic profile across the CD boundary (lower scheme). This potential well is populated by coherent molecules that experience a lower energy in the fundamental state than in the isolated, vapour state. The right panel depicts the profiles of the reduced field amplitude, <span class="html-italic">A</span>(<span class="html-italic">x</span>)/<span class="html-italic">A</span><sub>0</sub>, as a function of the radial parameter x, within the CD (reported from [<a href="#B38-ijms-25-03846" class="html-bibr">38</a>]). The centremost panel illustrates a single-CD decay profile, whereas the right panel shows the overlapping field amplitude between two adjacent CDs.</p>
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<p>The top panel represents the different ground states (vacua) experienced by water molecules (schematized as blue spheres) inside and outside the CD. Due to the em-field confinement, a potential gradient is established across the CD interface: the scheme serves to illustrate how the effective thermodynamic stability enjoyed by molecules belonging to the CD also depends on the radial position within the CD itself. Thus, the thermodynamic stability of coherence, which is usually derived from experimental data as different kinds/strengths/arrangements of “HB”, is the averaged result of many potential depths experienced by molecules depending on their radial position and on the width of the CD. What is considered to be a force existing between molecules (as generally conceived within the QM view of the first quantization), and which becomes macroscopically manifest as condensation when a critical density is reached, is now understood metaphorically to be exactly the same “apparent force” that we face if we tried to separate some marbles from each other resting at the bottom of the depression of an elastic cloth. This “force” is not something that exists intrinsically among the marbles. It is an emergent property that manifests itself through the coupling between the marbles and their (new) vacuum (ground) level, represented by the deformation of the cloth (second quantization). See the text for more details.</p>
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<p>Scheme of the qualitative change in the profile gradient at ground level for three different temperature values T<sub>1</sub> &lt; T<sub>2</sub> &lt; T<sub>3</sub> (grouped together in the upper panels (<b>a</b>–<b>c</b>)). The maximum depth of the well, predicted within the framework of theoretical QED, is independent of temperature, Δ<span class="html-italic"><sub>g</sub></span> ≈ 0.16 eV. The higher the thermal erosion and the narrower the energy cone-shaped well of the CD becomes (thermodynamically less stabilized), the more molecules (primarily at CD boundaries) experience weaker coherence. Increasing the temperature is comparable to reduce the available peripheral volume of the CD (making it narrower) while coherence is still strong (good potential depth). Panel (<b>d</b>) shows a sketch for 5 temperatures). A peculiar feature is that the profile of the energy-well becomes steeper and steeper as T increases, making it more difficult for molecules to “jump” out of the narrowing pit: such dynamics prevent an avalanche process that might otherwise lead to easy destruction of the coherent phase during evaporation. In fact, many CDs persist, in a reduced size, even in the gas phase, and to dissolve them completely, temperatures well above the thermodynamic boiling point of around 600 K must be reached [<a href="#B54-ijms-25-03846" class="html-bibr">54</a>].</p>
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