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Symmetry
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Particle Physics and Symmetry
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Particle Physics and Symmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: closed (30 November 2022) | Viewed by 16233

Special Issue Editor

Dr. Ayan Paul

E-Mail Website
Guest Editor
DESY, Hamburg & Humboldt Universität zu Berlin, Berlin, Germany
Interests: high energy physics; particle physics; flavour physics

Special Issue Information

Dear Colleagues,

Symmetries are one of the fundamental concepts that have guided the development of particle physics over several decades. On one hand, symmetries defined theoretical developments and identifications of interactions, and on the other hand, symmetries and their violation have been an important area of measurements in experimental particle physics. In this Special Issue, we gather the leading ideas about the application of symmetries in future theoretical development and how they will shape the experiments of the future. We would like to invite contributions from both experimentalists and theorists on the study and application of symmetries in particle physics and innovative ideas of physics beyond the Standard Model built on the foundations of extended symmetries. Ideas on how larger global and local symmetries can bring about novel dynamics, or on new sources for the breaking of discrete symmetries like CP or T are especially welcome, along with ideas on how to measure these dynamics.

Dr. Ayan Paul
Guest Editor

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Keywords

  • Symmetry
  • Gauge symmetries
  • Global symmetries
  • Anomalies
  • CP violation
  • New physics

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Published Papers (5 papers)

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18 pages, 572 KiB  
Article
Heavy Quark Expansion of Λb→Λ*(1520) Form Factors beyond Leading Order
by Marzia Bordone
Symmetry 2021, 13(4), 531; https://doi.org/10.3390/sym13040531 - 24 Mar 2021
Cited by 8 | Viewed by 1737
Abstract
I review the parametrisation of the full set of Λb→Λ*(1520) form factors in the framework of Heavy Quark Expansion, including next-to-leading-order O(αs) and, for the first time, next-to-leading-power [...] Read more.
I review the parametrisation of the full set of Λb→Λ*(1520) form factors in the framework of Heavy Quark Expansion, including next-to-leading-order O(αs) and, for the first time, next-to-leading-power O(1/mb) corrections. The unknown hadronic parameters are obtained by performing a fit to recent lattice QCD calculations. I investigate the compatibility of the Heavy Quark Expansion and the current lattice data, finding tension between these two approaches in the case of tensor and pseudo-tensor form factors, whose origin could come from an underestimation of the current lattice QCD uncertainties and higher order terms in the Heavy Quark Expansion. Full article
(This article belongs to the Special Issue Particle Physics and Symmetry)
►▼ Show Figures

Figure 1

Figure 1
<p>Comparison between the lattice results in Ref. [<a href="#B22-symmetry-13-00531" class="html-bibr">22</a>] (grey band) and the fit results for the HQE form factors (red band) for the vector and axial-vector form factors. The two bands represent the <math display="inline"><semantics> <mrow> <mn>68</mn> <mo>%</mo> </mrow> </semantics></math> interval.</p> Full article ">Figure 2
<p>Comparison between the lattice results in Ref. [<a href="#B22-symmetry-13-00531" class="html-bibr">22</a>] (grey band) and the predictions for tensor and pseudo-tensor form factors based on the fir results in <a href="#symmetry-13-00531-t001" class="html-table">Table 1</a> (red band). The two bands represent the <math display="inline"><semantics> <mrow> <mn>68</mn> <mo>%</mo> </mrow> </semantics></math> interval.</p> Full article ">Figure 3
<p>Comparison between the lattice QCD results in Ref. [<a href="#B22-symmetry-13-00531" class="html-bibr">22</a>] (grey band) and the predictions for tensor and pseudo-tensor form factors based on the fir results in <a href="#symmetry-13-00531-t001" class="html-table">Table 1</a> (red band) with lattice QCD uncertainties inflated of <math display="inline"><semantics> <mrow> <mn>20</mn> <mo>%</mo> </mrow> </semantics></math>. The two bands represent the <math display="inline"><semantics> <mrow> <mn>68</mn> <mo>%</mo> </mrow> </semantics></math> interval.</p> Full article ">
19 pages, 5624 KiB  
Article
Extending the Fully Bayesian Unfolding with Regularization Using a Combined Sampling Method
by Petr Baroň and Jiří Kvita
Symmetry 2020, 12(12), 2100; https://doi.org/10.3390/sym12122100 - 17 Dec 2020
Cited by 2 | Viewed by 2264
Abstract
Regularization extensions to the Fully Bayesian Unfolding are implemented and studied with an algorithm of combined sampling to find, in a reasonable computational time, an optimal value of the regularization strength parameter in order to obtain an unfolded result of a desired property, [...] Read more.
Regularization extensions to the Fully Bayesian Unfolding are implemented and studied with an algorithm of combined sampling to find, in a reasonable computational time, an optimal value of the regularization strength parameter in order to obtain an unfolded result of a desired property, like smoothness. Three regularization conditions using the curvature, entropy and derivatives are applied, as a model example, to several simulated spectra of top-pair quark pairs that are produced in high energy pp collisions. The existence of a minimum of a χ2 between the unfolded and particle-level spectra is discussed, with recommendations on the checks and validity of the usage of the regularization feature in Fully Bayesian Unfolding (FBU). Full article
(This article belongs to the Special Issue Particle Physics and Symmetry)
►▼ Show Figures

Figure 1

Figure 1
<p>Unfolding components of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math>: (<b>a</b>) particle spectra (green), pseudo data (blue), unfolding result <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold">T</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> (red), (<b>b</b>) efficiency <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> and acceptance <math display="inline"><semantics> <mi>η</mi> </semantics></math> corrections of statistically independent sets A and B, and (<b>c</b>) normalized migration matrix <math display="inline"><semantics> <msub> <mi>M</mi> <mi>A</mi> </msub> </semantics></math>.</p> Full article ">Figure 2
<p>(<b>a</b>) View of a part of the 16-dimensional log-likelihood <math display="inline"><semantics> <mrow> <mo form="prefix">log</mo> <mi>L</mi> <mo>(</mo> <mi mathvariant="bold">T</mi> <mo>)</mo> </mrow> </semantics></math> function as function of the 6th and 9th bin (<b>b</b>) normalized maximal values of <math display="inline"><semantics> <mrow> <mo form="prefix">log</mo> <mi>L</mi> <mo>(</mo> <msub> <mi>T</mi> <mn>6</mn> </msub> <mo>,</mo> <msub> <mi>T</mi> <mn>9</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, and (<b>c</b>) normalized maximal values of <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>(</mo> <msub> <mi>T</mi> <mn>6</mn> </msub> <mo>,</mo> <msub> <mi>T</mi> <mn>9</mn> </msub> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Marginalized 1D posteriors in the 6th (<b>a</b>) and 9th (<b>b</b>) bin of the <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> spectrum without regularization applied.</p> Full article ">Figure 4
<p>Unfolding the double-peaked <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> over-binned spectrum for different values of the regularization strength parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math>. The parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math> is normalized, such that <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <msub> <mi>τ</mi> <mi>rel</mi> </msub> </mrow> </semantics></math>, see <a href="#sec3-symmetry-12-02100" class="html-sec">Section 3</a>.</p> Full article ">Figure 5
<p>Relative <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> as function of the regularization strength parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math> and its minimum at <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>opt</mi> </msub> <mo>=</mo> <mn>2089</mn> </mrow> </semantics></math>. The parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math> is normalized, such that <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <msub> <mi>τ</mi> <mi>rel</mi> </msub> </mrow> </semantics></math>, see <a href="#sec3-symmetry-12-02100" class="html-sec">Section 3</a>. The vertical line represents the minimum of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The envelope of normalized regularization functions <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi mathvariant="bold">T</mi> <mo>)</mo> </mrow> </semantics></math> in the 6th and 9th bin. For sampling purposes, the gradient of <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>(</mo> <mi mathvariant="bold">T</mi> <mo>)</mo> <mo>−</mo> <mi>S</mi> <mo>(</mo> <mi mathvariant="bold">T</mi> <mo>)</mo> </mrow> </semantics></math> was used.</p> Full article ">Figure 7
<p>Posterior shifting and narrowing with increasing the regularization strength parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math> in a selected single bin: (<b>a</b>) no regularization applied <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>2089</mn> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>≈</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Mostly decreasing (<b>a</b>) curvature, (<b>b</b>) entropy, and (<b>c</b>) derivatives of the unfolded spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> with respect to <math display="inline"><semantics> <mi>τ</mi> </semantics></math>. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.</p> Full article ">Figure 9
<p>Cross-bin correlation matrix built from the correlation factor of likelihood, <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>(</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> </semantics></math> while using the curvature regularization for three different values of <math display="inline"><semantics> <mi>τ</mi> </semantics></math>.</p> Full article ">Figure 10
<p>The averaged cross bin correlations <math display="inline"><semantics> <msub> <mover accent="true"> <mi>C</mi> <mo>¯</mo> </mover> <mi>abs</mi> </msub> </semantics></math> (black) and <math display="inline"><semantics> <mover accent="true"> <mi>C</mi> <mo>¯</mo> </mover> </semantics></math> (pink) using (<b>a</b>) curvature, (<b>b</b>) entropy, and (<b>c</b>) derivative regularization for the <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> spectrum. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.</p> Full article ">Figure 11
<p>Variable <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> using (<b>a</b>) curvature (<b>b</b>) entropy and (<b>c</b>) derivative regularization illustrating the effect of spectra smoothing. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds. The red line indicates the minimal value of the <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Variable <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>denom</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> using (<b>a</b>) curvature, (<b>b</b>) entropy, and (<b>c</b>) derivative regularization illustrating the effect of narrowing the posteriors and decreasing the uncertainty. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.</p> Full article ">Figure 13
<p>Variables <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mo>/</mo> <msub> <mi>χ</mi> <mi>denom</mi> </msub> </mrow> </semantics></math> of the <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> spectrum using (<b>a</b>) curvature, (<b>b</b>) entropy, and (<b>c</b>) derivative regularization showing good correspondence. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.</p> Full article ">Figure 14
<p>Variable <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> using (<b>a</b>) curvature, (<b>b</b>) entropy, and (<b>c</b>) derivative regularization comparing combined (faster) sampling (blue) and full sampling (red). The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds. The vertical dotted lines indicate positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">Figure 15
<p>Variables (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>denom</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for the full sampling method; and (<b>c</b>) <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> using full (red) and combined (blue) sampling of the <math display="inline"><semantics> <msup> <mi>m</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> spectrum with an accidental minimum at <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>≈</mo> <mn>7000</mn> </mrow> </semantics></math> (curvature regularization). The vertical dotted lines indicate positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">Figure 16
<p>The result of unfolding (<b>a</b>) without regularization (<b>b</b>) with regularization and (<b>c</b>) with regularization applied only at second half of the spectrum <math display="inline"><semantics> <msup> <mi>m</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> while using the curvature in the case of a accidental minimum in <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> for one representative random seed.</p> Full article ">Figure 17
<p>Variables (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>denom</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for the full sampling method; and, (<b>c</b>) <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> using full (red) and combined (blue) sampling of the <math display="inline"><semantics> <msubsup> <mi>p</mi> <mrow> <mi>T</mi> </mrow> <mrow> <mi>t</mi> <mo>,</mo> <mi>had</mi> </mrow> </msubsup> </semantics></math> spectrum with a vanishing minimum (derivative regularization). The vertical dotted lines indicate positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">Figure 18
<p>Result of unfolding (<b>a</b>) without regularization and (<b>b</b>) with regularization of the spectrum <math display="inline"><semantics> <msubsup> <mi>p</mi> <mrow> <mi>T</mi> </mrow> <mrow> <mi>t</mi> <mo>,</mo> <mi>had</mi> </mrow> </msubsup> </semantics></math> using the derivatives in case of a hidden minimum in <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> for one representative random seed. Spectrum becomes smoother, but <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> does not improve due to the narrowing of posteriors.</p> Full article ">Figure 19
<p>Variables (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>denom</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for the full sampling method; and (<b>c</b>) <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> using full (red) and combined (blue) sampling of the <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> spectrum with the real minimum at <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>≈</mo> <mn>900</mn> </mrow> </semantics></math> (curvature regularization). The vertical dotted lines indicate positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">Figure 20
<p>The result of unfolding (<b>a</b>) without regularization and (<b>b</b>) with regularization of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> while using the curvature in case of a real minimum in <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> for one representative random seed.</p> Full article ">Figure 21
<p>Result of unfolding (<b>a</b>) without regularization and (<b>c</b>) with regularization of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>had</mi> </mrow> </msup> </semantics></math> using the curvature regularization for one representative random seed. Variable <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> (<b>b</b>) while using full (red) and combined (blue) sampling of the <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>had</mi> </mrow> </msup> </semantics></math> spectrum with minimum at <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1698</mn> </mrow> </semantics></math> (curvature regularization). In this case, the regularization is not needed. The vertical dotted lines indicate positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">Figure 22
<p>Result of unfolding (<b>a</b>) without regularization and (<b>c</b>) with regularization of the spectrum <math display="inline"><semantics> <msubsup> <mi>p</mi> <mi>T</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msubsup> </semantics></math> using entropy regularization for one representative random seed. Variable <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> (<b>b</b>) using full (red) and combined (blue) sampling of the <math display="inline"><semantics> <msubsup> <mi>p</mi> <mi>T</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msubsup> </semantics></math> spectrum with minimum at <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>2089</mn> </mrow> </semantics></math> (entropy regularization). The vertical dotted lines indicate the positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">
28 pages, 612 KiB  
Article
QCD Theory of the Hadrons and Filling the Yang–Mills Mass Gap
by Jay R. Yablon
Symmetry 2020, 12(11), 1887; https://doi.org/10.3390/sym12111887 - 16 Nov 2020
Viewed by 6498
Abstract
The rank-3 antisymmetric tensors which are the magnetic monopoles of SU(N) Yang–Mills gauge theory dynamics, unlike their counterparts in Maxwell’s U(1) electrodynamics, are non-vanishing, and do permit a net flux of Yang–Mills analogs to the magnetic field through closed spatial surfaces. When electric [...] Read more.
The rank-3 antisymmetric tensors which are the magnetic monopoles of SU(N) Yang–Mills gauge theory dynamics, unlike their counterparts in Maxwell’s U(1) electrodynamics, are non-vanishing, and do permit a net flux of Yang–Mills analogs to the magnetic field through closed spatial surfaces. When electric source currents of the same Yang–Mills dynamics are inverted and their fermions inserted into these Yang–Mills monopoles to create a system, this system in its unperturbed state contains exactly three fermions due to the monopole rank-3 and its three additive field strength gradient terms in covariant form. So to ensure that every fermion in this system occupies an exclusive quantum state, the Exclusion Principle is used to place each of the three fermions into the fundamental representation of the simple gauge group with an SU(3) symmetry. After the symmetry of the monopole is broken to make this system indivisible, the gauge bosons inside the monopole become massless, the SU(3) color symmetry of the fermions becomes exact, and a propagator is established for each fermion. The monopoles then have the same antisymmetric color singlet wavefunction as a baryon, and the field quanta of the magnetic fields fluxing through the monopole surface have the same symmetric color singlet wavefunction as a meson. Consequently, we are able to identify these fermions with colored quarks, the gauge bosons with gluons, the magnetic monopoles with baryons, and the fluxing entities with mesons, while establishing that the quarks and gluons remain confined and identifying the symmetry breaking with hadronization. Analytic tools developed along the way are then used to fill the Yang–Mills mass gap. Full article
(This article belongs to the Special Issue Particle Physics and Symmetry)

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17 pages, 1379 KiB  
Review
A Review of CP Violation Measurements in Charm at LHCb
by Federico Betti
Symmetry 2021, 13(8), 1482; https://doi.org/10.3390/sym13081482 - 12 Aug 2021
Cited by 1 | Viewed by 2218
Abstract
The LHCb experiment has been able to collect the largest sample ever produced of charm-hadron decays, performing a number of measurements of observables related to CP violation in the charm sector. In this document, the most recent results from LHCb on the [...] Read more.
The LHCb experiment has been able to collect the largest sample ever produced of charm-hadron decays, performing a number of measurements of observables related to CP violation in the charm sector. In this document, the most recent results from LHCb on the search of direct CP violation in D0→Ks0Ks0, D(s)+→h+π0 and D(s)+→h+η decays are summarised, in addition to the most precise measurement of time-dependent CP asymmetry in D0→h+h− decays and the first observation of mass difference between neutral charm-meson eigenstates. Full article
(This article belongs to the Special Issue Particle Physics and Symmetry)
►▼ Show Figures

Figure 1

Figure 1
<p>Distributions and fit projections of the <math display="inline"><semantics> <mrow> <mo mathvariant="normal">Δ</mo> <mi>m</mi> </mrow> </semantics></math> distribution for representative candidate categories (2017–2018 data sample).</p> Full article ">Figure 2
<p>Distribution of (<b>left</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>(</mo> <msup> <mi>h</mi> <mo>+</mo> </msup> <mi>η</mi> <mo>)</mo> </mrow> </semantics></math> and (<b>right</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>(</mo> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> <mi>γ</mi> <mo>)</mo> </mrow> </semantics></math> for (<b>top</b>) <math display="inline"><semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>+</mo> </msubsup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mi>η</mi> </mrow> </semantics></math> and (<b>bottom</b>) <math display="inline"><semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>+</mo> </msubsup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <mi>η</mi> </mrow> </semantics></math> candidates, summed over all categories of the simultaneous fit, with projections of the fit result overlaid. <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>h</mi> <mo>+</mo> </msup> <mi>η</mi> </mrow> </semantics></math> contribution is shown in dashed red, <math display="inline"><semantics> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <mo>→</mo> <msup> <mi>h</mi> <mo>+</mo> </msup> <mi>η</mi> </mrow> </semantics></math> in solid grey, pure combinatorial decays in dashed black and partially reconstructed background in dotted magenta. The misidentification background is too small to be seen.</p> Full article ">Figure 3
<p>Distribution of (<b>left</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>(</mo> <msup> <mi>h</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mn>0</mn> </msup> <mo>)</mo> </mrow> </semantics></math> and (<b>right</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>(</mo> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> <mi>γ</mi> <mo>)</mo> </mrow> </semantics></math> for (<b>top</b>) <math display="inline"><semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>+</mo> </msubsup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mn>0</mn> </msup> </mrow> </semantics></math> and (<b>bottom</b>) <math display="inline"><semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>+</mo> </msubsup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mn>0</mn> </msup> </mrow> </semantics></math> candidates, summed over all categories of the simultaneous fit, with projections of the fit result overlaid. <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>h</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mn>0</mn> </msup> </mrow> </semantics></math> contribution is shown in dashed red, <math display="inline"><semantics> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <mo>→</mo> <msup> <mi>h</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mn>0</mn> </msup> </mrow> </semantics></math> in solid grey, pure combinatorial decays in dashed black and real-<math display="inline"><semantics> <msup> <mi>π</mi> <mn>0</mn> </msup> </semantics></math> combinatorial background in dotted green. The misidentification background is too small to be seen.</p> Full article ">Figure 4
<p>Distribution of <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>(</mo> <msup> <mi>D</mi> <mn>0</mn> </msup> <msup> <mi>π</mi> <mo>+</mo> </msup> <mo>)</mo> </mrow> </semantics></math> for (<b>left</b>) <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>−</mo> </msup> <msup> <mi>π</mi> <mo>+</mo> </msup> </mrow> </semantics></math>, (<b>center</b>) <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <msup> <mi>K</mi> <mo>−</mo> </msup> </mrow> </semantics></math> and (<b>right</b>) <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>−</mo> </msup> </mrow> </semantics></math> candidates. The signal window and the lateral window used to remove the combinatorial background (grey filled area) are delimited by the vertical dashed lines. Fit projections are overlaid.</p> Full article ">Figure 5
<p>(<b>Left</b>) Normalised distributions of the <math display="inline"><semantics> <msup> <mi>D</mi> <mn>0</mn> </msup> </semantics></math> transverse momentum, in different colours for each decay-time interval. Decay time increases from blue to yellow colour. (<b>Right</b>) Linear fit to the time-dependent asymmetry (red) before and (black) after the kinematic weighting. The plots correspond to the <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>−</mo> </msup> <msup> <mi>π</mi> <mo>+</mo> </msup> </mrow> </semantics></math> candidates recorded in 2016 with the magnet polarity pointing upwards.</p> Full article ">Figure 6
<p>Asymmetry as a function of <math display="inline"><semantics> <msup> <mi>D</mi> <mn>0</mn> </msup> </semantics></math> decay time, for (<b>left</b>) <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <msup> <mi>K</mi> <mo>−</mo> </msup> </mrow> </semantics></math> and (<b>right</b>) <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>−</mo> </msup> </mrow> </semantics></math> samples, with linear fit superimposed.</p> Full article ">Figure 7
<p>Binning scheme of the <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>→</mo> <msubsup> <mi>K</mi> <mi>s</mi> <mn>0</mn> </msubsup> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>−</mo> </msup> </mrow> </semantics></math> Dalitz plot.</p> Full article ">Figure 8
<p>(<b>Left</b>) <math display="inline"><semantics> <mrow> <mi>C</mi> <mspace width="-0.166667em"/> <mi>P</mi> </mrow> </semantics></math>-averaged yield ratios and (<b>right</b>) differences of <math display="inline"><semantics> <msup> <mi>D</mi> <mn>0</mn> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mn>0</mn> </msup> </semantics></math> yield ratios as a function of <math display="inline"><semantics> <msup> <mi>D</mi> <mn>0</mn> </msup> </semantics></math> decay time, shown for each Dalitz-plot bin. Fit projections are overlaid.</p> Full article ">
15 pages, 4186 KiB  
Review
The International Linear Collider Project—Its Physics and Status
by Hitoshi Yamamoto
Symmetry 2021, 13(4), 674; https://doi.org/10.3390/sym13040674 - 13 Apr 2021
Cited by 8 | Viewed by 2392
Abstract
The discovery of Higgs particle has ushered in a new era of particle physics. Even though the list of members of the standard theory of particle physics is now complete, the shortcomings of the theory became ever more acute. It is generally considered [...] Read more.
The discovery of Higgs particle has ushered in a new era of particle physics. Even though the list of members of the standard theory of particle physics is now complete, the shortcomings of the theory became ever more acute. It is generally considered that the best solution to the problems is an electron–positron collider that can study Higgs particle with high precision and high sensitivity; namely, a Higgs factory. Among a few candidates for Higgs factory, the International Linear Collider (ILC) is currently the most advanced in its program. In this article, we review the physics and the project status of the ILC including its energy expandability. Full article
(This article belongs to the Special Issue Particle Physics and Symmetry)
►▼ Show Figures

Figure 1

Figure 1
<p>A schematic drawing of the International Linear Collider (ILC) (not to scale).</p> Full article ">Figure 2
<p>A 1.3 GHz 9-cell superconducting cavities for the ILC. The length is 1.25 m.</p> Full article ">Figure 3
<p>Maximum gradient and yield of ‘as received’ cavities produced for the European X-ray Free Electron Laser (E-XFEL) by two vendors [<a href="#B11-symmetry-13-00674" class="html-bibr">11</a>]. Vendor RI employs a production process that closely follows the ILC specification.</p> Full article ">Figure 4
<p>The “S1-Global” project in which parts manufactured at different locations all over the world are assembled together with “plug-compatible” designs.</p> Full article ">Figure 5
<p>The achieved vertical beam size at ATF2 facility [<a href="#B9-symmetry-13-00674" class="html-bibr">9</a>]. The latest value of 41 nm is within 10% of the goal of 37 nm.</p> Full article ">Figure 6
<p>The luminosity vs. center-of-mass energy for the four candidates for electron-positron Higgs factory. The luminosity is for single interaction point while the effect of polarization is not included ([<a href="#B9-symmetry-13-00674" class="html-bibr">9</a>], and references therein).</p> Full article ">Figure 7
<p>(<b>a</b>) The processes to produce Higgs particles at the ILC Higgs factory [<a href="#B4-symmetry-13-00674" class="html-bibr">4</a>]. (<b>b</b>) Reconstruction of Higgs particle with the recoil mass technique at collision energy of 250 GeV [<a href="#B12-symmetry-13-00674" class="html-bibr">12</a>].</p> Full article ">Figure 8
<p>Improvements by adding ILC data to the upgraded LHC data [<a href="#B9-symmetry-13-00674" class="html-bibr">9</a>]. The all analyses assume that there are no Higgs decay final states other than those of the Standard Theory and that the forms of <span class="html-italic">HZ</span> and <span class="html-italic">HW</span> couplings are the same as those of the Standard Theory. For each bar, the light (dark) color corresponds to an optimistic (conservative) projection.</p> Full article ">Figure 9
<p>Relative precisions obtainable by the data with and without polarizations [<a href="#B9-symmetry-13-00674" class="html-bibr">9</a>]. The results of the ILC are combined with those of upgraded LHC (HL-LHC). The polarizations are assumed to be ±80% for electrons and ±30% for positrons, and the data is divided to (−+, +−, ++, −−) = (45%, 45%, 5%, 5%).</p> Full article ">Figure 10
<p>Separation powers of the ILC Higgs factory for 9 new theoretical models beyond the Standard Theory that are considered unlikely to be rejected by the upgraded LHC [<a href="#B19-symmetry-13-00674" class="html-bibr">19</a>].</p> Full article ">Figure 11
<p>An example for pair creation of new particle that decays to the dark matter candidate and a muon: <math display="inline"><semantics> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> <mo>→</mo> <msup> <mover accent="true"> <mi>μ</mi> <mo stretchy="false">˜</mo> </mover> <mo>+</mo> </msup> <msup> <mover accent="true"> <mi>μ</mi> <mo stretchy="false">˜</mo> </mover> <mo>−</mo> </msup> <mo>,</mo> <mo> </mo> <mover accent="true"> <mi>μ</mi> <mo stretchy="false">˜</mo> </mover> <mo>→</mo> <mi>χ</mi> <mi>μ</mi> </mrow> </semantics></math> [<a href="#B9-symmetry-13-00674" class="html-bibr">9</a>]. The regions for 5σ discovery and 95% exclusion are shown. The analysis corresponds to 500 fb<sup>−1</sup> of data taken at collision energy of 500 GeV. The signature for 250 GeV collision energy is essentially the same.</p> Full article ">Figure 12
<p>At the threshold of top pair production, the top mass and decay rate can be measured with high precision [<a href="#B22-symmetry-13-00674" class="html-bibr">22</a>].</p> Full article ">Figure 13
<p>The right–handed and left–handed top coupling to the Z are shown for the Standard Theory and other new theories. The ILC precision is shown as the dotted red oval near the center [<a href="#B24-symmetry-13-00674" class="html-bibr">24</a>].</p> Full article ">
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