String geometry theory is one of the candidates of nonperturbative formulation of string theory. In this paper, we have shown that dimensionally reduced string geometry theories have what we call <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M2"><ns0:mi>T</ns0:mi></ns0:math>-symmetry. In the case of the dimensional reduction in space-like directions, the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M3"><ns0:mi>T</ns0:mi></ns0:math>-symmetry transformation gives the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M4"><ns0:mi>T</ns0:mi></ns0:math>-dual transformation between the Type IIA and IIB perturbative vacua. In the case of the dimensional reduction in the direction of string geometry time <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M5"><ns0:mover accent="true"><ns0:mi>&#964;</ns0:mi><ns0:mo stretchy="true">&#175;</ns0:mo></ns0:mover></ns0:math>, the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M6"><ns0:mi>T</ns0:mi></ns0:math>-symmetry transformation is independent of the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M7"><ns0:mi>T</ns0:mi></ns0:math>-dual transformation and gives a symmetry that cannot be seen in the perturbative string theories.

Japan

<ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M1"><ns0:mi>T</ns0:mi></ns0:math>-Symmetry in String Geometry Theory

The implication of the neutrino mass model based on <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M3"><ns0:mi>&#916;</ns0:mi><ns0:mfenced open="(" close=")"><ns0:mrow><ns0:mn>27</ns0:mn></ns0:mrow></ns0:mfenced></ns0:math> discrete flavor symmetry, on parameters of neutrino oscillations, CP violation, and effective neutrino masses is studied using the Type I seesaw mechanism. The standard model particle content is extended by adding two additional Higgs doublets, three right-handed neutrinos, and two scalar triplets under <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M4"><ns0:mi>&#916;</ns0:mi><ns0:mfenced open="(" close=")"><ns0:mrow><ns0:mn>27</ns0:mn></ns0:mrow></ns0:mfenced></ns0:math> symmetry predicting diagonal charged lepton mass matrix. This can generate the desired deviation from <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M5"><ns0:mi>&#956;</ns0:mi><ns0:mo>&#8208;</ns0:mo><ns0:mi>&#964;</ns0:mi></ns0:math> symmetry. The resulting neutrino oscillation parameters are well agreed with the latest global fit oscillation data. The sum of the three absolute neutrino mass eigenvalues, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M6"><ns0:msub><ns0:mrow><ns0:mo>&#8721;</ns0:mo></ns0:mrow><ns0:mrow><ns0:mi>i</ns0:mi></ns0:mrow></ns0:msub><ns0:mfenced open="|" close="|"><ns0:mrow><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>i</ns0:mi></ns0:mrow></ns0:msub></ns0:mrow></ns0:mfenced></ns0:math> (<ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M7"><ns0:mi>i</ns0:mi><ns0:mo>=</ns0:mo><ns0:mn>1</ns0:mn><ns0:mo>,</ns0:mo><ns0:mn>2</ns0:mn><ns0:mo>,</ns0:mo><ns0:mn>3</ns0:mn></ns0:math>), is found to be consistent with the value given by the latest Planck cosmological data, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M8"><ns0:msub><ns0:mrow><ns0:mo>&#8721;</ns0:mo></ns0:mrow><ns0:mrow><ns0:mi>i</ns0:mi></ns0:mrow></ns0:msub><ns0:mfenced open="|" close="|"><ns0:mrow><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>i</ns0:mi></ns0:mrow></ns0:msub></ns0:mrow></ns0:mfenced><ns0:mo>&lt;</ns0:mo><ns0:mn>0.12</ns0:mn></ns0:math>&#8201;eV. The model further predicts effective neutrino masses for neutrinoless double beta decay, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M9"><ns0:mn>4.15</ns0:mn><ns0:mtext>&#8201;</ns0:mtext><ns0:mtext>meV</ns0:mtext><ns0:mo>&#8804;</ns0:mo><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>e</ns0:mi><ns0:mi>e</ns0:mi></ns0:mrow></ns0:msub><ns0:mo>&#8804;</ns0:mo><ns0:mn>30.6</ns0:mn></ns0:math>&#8201;meV; tritium beta decay, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M10"><ns0:mn>8.4</ns0:mn><ns0:mtext>&#8201;</ns0:mtext><ns0:mtext>meV</ns0:mtext><ns0:mo>&#8804;</ns0:mo><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>&#946;</ns0:mi></ns0:mrow></ns0:msub><ns0:mo>&#8804;</ns0:mo><ns0:mn>30.5</ns0:mn></ns0:math>&#8201;meV; Jarlskog invariant, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M11"><ns0:msub><ns0:mrow><ns0:mi>J</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>C</ns0:mi><ns0:mi>P</ns0:mi></ns0:mrow></ns0:msub><ns0:mo>=</ns0:mo><ns0:mo>&#177;</ns0:mo><ns0:mn>0.022</ns0:mn></ns0:math> for CP violation; baryon asymmetry <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M12"><ns0:msub><ns0:mrow><ns0:mi>Y</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>B</ns0:mi></ns0:mrow></ns0:msub><ns0:mo>=</ns0:mo><ns0:mn>1.15</ns0:mn><ns0:mo>&#215;</ns0:mo><ns0:mn>1</ns0:mn><ns0:msup><ns0:mrow><ns0:mn>0</ns0:mn></ns0:mrow><ns0:mrow><ns0:mo>&#8722;</ns0:mo><ns0:mn>10</ns0:mn></ns0:mrow></ns0:msup></ns0:math> for normal hierarchical case; and also <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M13"><ns0:mn>49.5</ns0:mn><ns0:mtext>&#8201;</ns0:mtext><ns0:mtext>meV</ns0:mtext><ns0:mo>&#8804;</ns0:mo><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>e</ns0:mi><ns0:mi>e</ns0:mi></ns0:mrow></ns0:msub><ns0:mo>&#8804;</ns0:mo><ns0:mn>51.7</ns0:mn></ns0:math>&#8201;meV, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M14"><ns0:mn>49.5</ns0:mn><ns0:mtext>&#8201;</ns0:mtext><ns0:mtext>meV</ns0:mtext><ns0:mo>&#8804;</ns0:mo><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>&#946;</ns0:mi></ns0:mrow></ns0:msub><ns0:mo>&#8804;</ns0:mo><ns0:mn>51.4</ns0:mn></ns0:math>&#8201;meV, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M15"><ns0:msub><ns0:mrow><ns0:mi>J</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>C</ns0:mi><ns0:mi>P</ns0:mi></ns0:mrow></ns0:msub><ns0:mo>=</ns0:mo><ns0:mo>&#177;</ns0:mo><ns0:mn>0.022</ns0:mn></ns0:math>, and <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M16"><ns0:msub><ns0:mrow><ns0:mi>Y</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>B</ns0:mi></ns0:mrow></ns0:msub><ns0:mo>=</ns0:mo><ns0:mn>1.12</ns0:mn><ns0:mo>&#215;</ns0:mo><ns0:mn>1</ns0:mn><ns0:msup><ns0:mrow><ns0:mn>0</ns0:mn></ns0:mrow><ns0:mrow><ns0:mo>&#8722;</ns0:mo><ns0:mn>10</ns0:mn></ns0:mrow></ns0:msup></ns0:math> for inverted hierarchical case, respectively.

India

Deviations From <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M1"><ns0:mi>&#956;</ns0:mi><ns0:mo>&#8208;</ns0:mo><ns0:mi>&#964;</ns0:mi></ns0:math> Symmetry Using <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M2"><ns0:mi>&#916;</ns0:mi><ns0:mfenced open="(" close=")"><ns0:mrow><ns0:mn>27</ns0:mn></ns0:mrow></ns0:mfenced></ns0:math> Group on Neutrino Masses and Mixings

This work presents the application of the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M2"><ns0:mi>q</ns0:mi></ns0:math>-algebra in the glueball spectrum. This algebra is implemented through Jackson&#8217;s derivatives in a Schr&#246;dinger-like equation resulting from the gravity fluctuations around the braneworld scenario in five dimensions. In our prescription, we consider a four-dimensional AdS4 brane, living in AdS5 bulk, that is also known for describing locally localized gravity via quasizero mode. At the appropriate limit, this background leads to confinement and allows us to find the shape of the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M3"><ns0:mi>q</ns0:mi></ns0:math>-deformed glueball spectrum in the AdS5/QCD4 correspondence. The introduction of the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M4"><ns0:mi>q</ns0:mi></ns0:math>-deformation provides us with a richer glueball spectrum.

Brazil

<ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M1"><ns0:mi>q</ns0:mi></ns0:math>-Deformed Glueball Spectrum in AdS/QCD Correspondence

A Hermitian matrix can be parametrized by a set consisting of its determinant and the eigenvalues of its submatrices. We established a group of equations which connect these variables with the mixing parameters of diagonalization. These equations are simple in structure and manifestly invariant in form under the symmetry operations of dilatation, translation, rephasing, and permutation. When applied to the problem of neutrino oscillation in matter, they produced two new “matter invariants” which are confirmed by available data.

Taiwan

United States

Hermitian Matrix Diagonalization and Its Symmetry Properties

This study addressed the scalar field quasinormal ringing behavior of black holes. We investigated scalar field perturbations in Bardeen black hole spacetime in 5-dimensional Einstein–Gauss–Bonnet (EGB) gravity. Using the sixth-order Padé approximation and the finite difference method, we computed the frequency of quasinormal modes (QNMs) in the spacetime background. The calculations demonstrated that the real part of the QNMs <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M1"><ns0:mi>&#969;</ns0:mi></ns0:math> increased, whereas the imaginary part decreased with increase in the magnetic charge parameter <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M2"><ns0:mi>Q</ns0:mi></ns0:math> of the Bardeen black hole for a fixed Gauss&#8211;Bonnet parameter <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M3"><ns0:mi>&#945;</ns0:mi></ns0:math>. This was also valid when <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M4"><ns0:mi>Q</ns0:mi></ns0:math> was fixed and <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M5"><ns0:mi>&#945;</ns0:mi></ns0:math> increased, where in the real part of the QNMs increased and the absolute value of the imaginary part decreased. However, the change in the latter case was more significant than that in the former; thus, the frequency of eigenvibration of this black hole background under the scalar field perturbation increased and the decay of eigenvibration decreased with increase in <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M6"><ns0:mi>&#945;</ns0:mi></ns0:math> or <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M7"><ns0:msup><ns0:mrow><ns0:mi>Q</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:math>. Moreover, this result shows that the effect of <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M8"><ns0:mi>&#945;</ns0:mi></ns0:math> on the intrinsic vibration of this black hole was greater than that of <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M9"><ns0:mi>Q</ns0:mi></ns0:math>.

China

Quasinormal Modes of Bardeen Black Hole in 5-Dimensional Gauss–Bonnet Gravity

Information theory has proven to be a worthwhile tool for investigating the implications of the Higgs sector in the Next-to-Minimal Supersymmetric Standard Model (NMSSM) using Higgs information at the Large Hadron Collider (LHC) assessed through the entropy constructed by means of the branching ratios of decay channels of the Higgs boson. The present article focuses on the parameter space of supersymmetric extension with an extra term of singlet in light of various experimental constraints. Our findings show the most preferred values of <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M1"><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>0</ns0:mn></ns0:mrow></ns0:msub></ns0:math>, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M2"><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>1</ns0:mn><ns0:mo>/</ns0:mo><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msub></ns0:math>, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M3"><ns0:msub><ns0:mrow><ns0:mi>A</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>0</ns0:mn></ns0:mrow></ns0:msub></ns0:math>, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M4"><ns0:mi mathvariant="normal">tan</ns0:mi><ns0:mi>&#946;</ns0:mi></ns0:math>, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M5"><ns0:mi>&#955;</ns0:mi></ns0:math>, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M6"><ns0:msub><ns0:mrow><ns0:mi>&#956;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mtext>eff</ns0:mtext></ns0:mrow></ns0:msub></ns0:math>, neutralino lightest supersymmetric particle (LSP) <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M7"><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:msubsup><ns0:mrow><ns0:mover accent="true"><ns0:mi>&#967;</ns0:mi><ns0:mo stretchy="true">~</ns0:mo></ns0:mover></ns0:mrow><ns0:mrow><ns0:mn>1</ns0:mn></ns0:mrow><ns0:mrow><ns0:mn>0</ns0:mn></ns0:mrow></ns0:msubsup></ns0:mrow></ns0:msub></ns0:math>, lighter chargino <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M8"><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:msubsup><ns0:mrow><ns0:mover accent="true"><ns0:mi>&#967;</ns0:mi><ns0:mo stretchy="true">~</ns0:mo></ns0:mover></ns0:mrow><ns0:mrow><ns0:mn>1</ns0:mn></ns0:mrow><ns0:mrow><ns0:mo>&#177;</ns0:mo></ns0:mrow></ns0:msubsup></ns0:mrow></ns0:msub></ns0:math>, singlino <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M9"><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:msubsup><ns0:mrow><ns0:mover accent="true"><ns0:mi>&#967;</ns0:mi><ns0:mo stretchy="true">~</ns0:mo></ns0:mover></ns0:mrow><ns0:mrow><ns0:mn>5</ns0:mn></ns0:mrow><ns0:mrow><ns0:mn>0</ns0:mn></ns0:mrow></ns0:msubsup></ns0:mrow></ns0:msub></ns0:math>, and gluino <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M10"><ns0:msub><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mover accent="true"><ns0:mi>g</ns0:mi><ns0:mo stretchy="true">~</ns0:mo></ns0:mover></ns0:mrow></ns0:msub></ns0:math> to be around 1.93&#8201;TeV, 1.78&#8201;TeV, &#8722;3.62&#8201;TeV, 27.5, 0.012, 665.7&#8201;GeV, 740&#8201;GeV, 790&#8201;GeV, 11.24&#8201;TeV, and 3.70&#8201;TeV, respectively, that is compatible with the relic density of dark matter.

Higgs Information and NMSSM at the Large Hadron Collider

For the static spherically symmetric dilatonic black hole described by the Gibbons–Maeda–Garfinkle–Horowitz–Strominger geometry, we analyze the timelike trajectories for electrically charged test particles. Both cases of an electric black hole and a magnetic one are considered. Finally, we are obtaining the solution to the Klein–Gordon equation in terms of Heun confluent functions and the corresponding energy spectrum. A special attention is given to the role of the dilaton parameter.

Romania

Charged Particles Moving around a Spherically Symmetric Dilatonic Black Hole

There is growing interest in the development of a muon collider that would make it possible to produce lepton collisions at energies of several TeV. Among others, there can be significant contributions to electroweak gauge boson, Higgs boson, and top quark physics. In this work, we pay attention to the latter, in particular, effective flavor-changing (FC) top-quark interactions. We discuss the flavor-changing <math id="M1"><mi>t</mi><mover><mi>q</mi><mo>¯</mo></mover><mfenced><mrow><mi>q</mi><mo>=</mo><mi>u</mi><mo>,</mo><mi>c</mi></mrow></mfenced></math> production processes that can be a good probe of the dimension-six top quark four-fermion and fermion-boson operators in the SMEFT. We consider all sixteen operators that can generate flavor-changing top quark couplings. After comparing with the current LHC bounds, we find potential limits three or four orders of magnitude stronger for four-fermion operators. Concerning fermion-boson couplings, for the tensor operators <math id="M2"><msub><mrow><mi>Q</mi></mrow><mrow><mi>u</mi><mi>W</mi></mrow></msub></math> and <math id="M3"><msub><mrow><mi>Q</mi></mrow><mrow><mi>u</mi><mi>B</mi></mrow></msub></math>, we obtain the highest sensitivity. We also observe that the effective <math id="M4"><mi>W</mi></math> approximation (EWA) does not apply with <math id="M5"><msub><mrow><mi>Q</mi></mrow><mrow><mi>u</mi><mi>B</mi></mrow></msub></math>.

Mexico

Top Quark Flavor Changing Couplings at a Muon Collider

The neutrino mass hierarchy determination (<ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M2"><ns0:mi>&#957;</ns0:mi></ns0:math> MHD) is one of the main goals of the major current and future neutrino experiments. The statistical analysis usually proceeds from a standard method, a single-dimensional estimator <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M3"><ns0:mfenced open="(" close=")"><ns0:mrow><ns0:mn>1</ns0:mn><ns0:mtext>D</ns0:mtext><ns0:mo>&#8722;</ns0:mo><ns0:mi>&#916;</ns0:mi><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:mrow></ns0:mfenced></ns0:math> that shows some drawbacks and concerns, together with a debatable strategy. The drawbacks and considerations of the standard method will be explained through the following three main issues. The first issue corresponds to the limited power of the standard method. The <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M4"><ns0:mi>&#916;</ns0:mi><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:math> estimator provides us with different results when different simulation procedures were used. Regarding the second issue, when <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M5"><ns0:msubsup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi mathvariant="normal">min</ns0:mi><ns0:mfenced open="(" close=")"><ns0:mrow><ns0:mtext>NH</ns0:mtext></ns0:mrow></ns0:mfenced></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msubsup></ns0:math> and <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M6"><ns0:msubsup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi mathvariant="normal">min</ns0:mi><ns0:mfenced open="(" close=")"><ns0:mrow><ns0:mtext>IH</ns0:mtext></ns0:mrow></ns0:mfenced></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msubsup></ns0:math> are drawn in a 2D map, their strong positive correlation manifests <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M7"><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:math> as a bidimensional variable, instead of a single-dimensional estimator. The overlapping between the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M8"><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:math> distributions of the two hypotheses leads to an experiment sensitivity reduction. The third issue corresponds to the robustness of the standard method. When the JUNO sensitivity is obtained using different procedures, either with <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M9"><ns0:mi>&#916;</ns0:mi><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:math> as one-dimensional or <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M10"><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:math> as two-dimensional estimator, the experimental sensitivity varies with the different values of the atmospheric mass, the input parameter. We computed the oscillation of <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M11"><ns0:mfenced open="|" close="|"><ns0:mrow><ns0:mover accent="false"><ns0:mrow><ns0:mi>&#916;</ns0:mi><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:mrow><ns0:mrow><ns0:mo>&#175;</ns0:mo></ns0:mrow></ns0:mover></ns0:mrow></ns0:mfenced></ns0:math> with the input parameter values, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M12"><ns0:msub><ns0:mrow><ns0:mfenced open="|" close="|"><ns0:mrow><ns0:mi>&#916;</ns0:mi><ns0:msup><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:mrow></ns0:mfenced></ns0:mrow><ns0:mrow><ns0:mtext>input</ns0:mtext></ns0:mrow></ns0:msub></ns0:math>. The MH significance using the standard method, <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M13"><ns0:mi>&#916;</ns0:mi><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:math>, strongly depends on the values of the parameter <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M14"><ns0:msub><ns0:mrow><ns0:mfenced open="|" close="|"><ns0:mrow><ns0:mi>&#916;</ns0:mi><ns0:msup><ns0:mrow><ns0:mi>m</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:mrow></ns0:mfenced></ns0:mrow><ns0:mrow><ns0:mtext>input</ns0:mtext></ns0:mrow></ns0:msub></ns0:math>. Consequently, the experiment sensitivity depends on the precision of the atmospheric mass. This evaluation of the standard method confirms the drawbacks.

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Statistical Issues on the Neutrino Mass Hierarchy with <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M1"><ns0:mi>&#916;</ns0:mi><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:math>

The measurements of the total cross section of the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M2"><ns0:msup><ns0:mrow><ns0:mi>e</ns0:mi></ns0:mrow><ns0:mrow><ns0:mo>+</ns0:mo></ns0:mrow></ns0:msup><ns0:msup><ns0:mrow><ns0:mi>e</ns0:mi></ns0:mrow><ns0:mrow><ns0:mo>&#8722;</ns0:mo></ns0:mrow></ns0:msup><ns0:mo>&#10230;</ns0:mo><ns0:mi>&#947;</ns0:mi><ns0:mi>&#947;</ns0:mi><ns0:mfenced open="(" close=")"><ns0:mrow><ns0:mi>&#947;</ns0:mi></ns0:mrow></ns0:mfenced></ns0:math> reaction from the VENUS, TOPAS, OPAL, DELPHI, ALEPH, and L3 collaborations, collected between 1989 and 2003, are used to perform a <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M3"><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:math> test to validate the current quantum electrodynamics (QED) theory and search for possible deviations with the direct contact term annihilation. By observing a deviation from the QED predictions on the total cross section of the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M4"><ns0:msup><ns0:mrow><ns0:mi>e</ns0:mi></ns0:mrow><ns0:mrow><ns0:mo>+</ns0:mo></ns0:mrow></ns0:msup><ns0:msup><ns0:mrow><ns0:mi>e</ns0:mi></ns0:mrow><ns0:mrow><ns0:mo>&#8722;</ns0:mo></ns0:mrow></ns0:msup><ns0:mo>&#10230;</ns0:mo><ns0:mi>&#947;</ns0:mi><ns0:mi>&#947;</ns0:mi><ns0:mfenced open="(" close=")"><ns0:mrow><ns0:mi>&#947;</ns0:mi></ns0:mrow></ns0:mfenced></ns0:math> reaction above <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M5"><ns0:msqrt><ns0:mrow><ns0:mi>s</ns0:mi></ns0:mrow></ns0:msqrt><ns0:mo>=</ns0:mo><ns0:mn>180.0</ns0:mn></ns0:math>&#8201;GeV, a non-QED direct contact term is introduced following the dimension 6 effective theory to explain the deviation. In the non-QED direct contact term, a threshold energy scale <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M6"><ns0:mi>&#923;</ns0:mi></ns0:math> is included and explained to the finite interaction length in direct contact term and in consequence the size of the electron involved in the annihilation area. The experimental data of the total cross section is compared to the QED cross section by a <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M7"><ns0:msup><ns0:mrow><ns0:mi>&#967;</ns0:mi></ns0:mrow><ns0:mrow><ns0:mn>2</ns0:mn></ns0:mrow></ns0:msup></ns0:math> test, which gives a best fit of the <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M8"><ns0:mi>&#923;</ns0:mi></ns0:math> to be <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M9"><ns0:mn>1576</ns0:mn><ns0:mo>&#177;</ns0:mo><ns0:mn>202</ns0:mn></ns0:math>&#8201;GeV, corresponding to a finite interaction length of <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M10"><ns0:msub><ns0:mrow><ns0:mi>r</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>e</ns0:mi></ns0:mrow></ns0:msub><ns0:mo>=</ns0:mo><ns0:mfenced open="(" close=")"><ns0:mrow><ns0:mn>1.25</ns0:mn><ns0:mo>&#177;</ns0:mo><ns0:mn>0.16</ns0:mn></ns0:mrow></ns0:mfenced><ns0:mo>&#215;</ns0:mo><ns0:mn>1</ns0:mn><ns0:msup><ns0:mrow><ns0:mn>0</ns0:mn></ns0:mrow><ns0:mrow><ns0:mo>&#8722;</ns0:mo><ns0:mn>17</ns0:mn></ns0:mrow></ns0:msup></ns0:math> (cm). In the direct contact term annihilation, this interaction length is a measure of the size of an electron <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M11"><ns0:msub><ns0:mrow><ns0:mi>r</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>e</ns0:mi></ns0:mrow></ns0:msub></ns0:math>. By combining all the data results from the mentioned collaborations, we have at least 2 to 3 times more statistics than every single experiment at high <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M12"><ns0:msqrt><ns0:mrow><ns0:mi>s</ns0:mi></ns0:mrow></ns0:msqrt></ns0:math> region. This induces the best precision on <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M13"><ns0:msub><ns0:mrow><ns0:mi>r</ns0:mi></ns0:mrow><ns0:mrow><ns0:mi>e</ns0:mi></ns0:mrow></ns0:msub></ns0:math> compared to the previous measurements.

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Hint for a Minimal Interaction Length in <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML" id="M1"><ns0:msup><ns0:mrow><ns0:mi>e</ns0:mi></ns0:mrow><ns0:mrow><ns0:mo>+</ns0:mo></ns0:mrow></ns0:msup><ns0:msup><ns0:mrow><ns0:mi>e</ns0:mi></ns0:mrow><ns0:mrow><ns0:mo>&#8722;</ns0:mo></ns0:mrow></ns0:msup><ns0:mo>&#10230;</ns0:mo><ns0:mi>&#947;</ns0:mi><ns0:mi>&#947;</ns0:mi></ns0:math> Annihilation in Total Cross Section of Center-of-Mass Energies 55-207&#8201;GeV