High Energy Physics
A topical collection in Particles (ISSN 2571-712X).
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Dear Colleagues,
In 1948, Dr. Phyllis Freier and his team discovered the relativistic heavy ions component of primary cosmic rays. a new field in nuclear and particle physics was created. Twenty-two years later, in August 1970, the first heavy ions beam was used at the JINR Dubna to evidence the cumulative negative ion production in DC collisions at 5 A GeV, using the Synchrophasotron U-10.
In the fifty years of relativistic and ultra-relativistic heavy ion physics using accelerator systems, many important results and discoveries have been reported, including those related to quark–gluon plasma formation and with the Big Bang cosmological scenario.
Therefore, the goal of the Editorial Board of the Particles journal is to dedicate a collection from this very interesting and dynamic field to present significant experimental results; new experiments and their goals; new theoretical ways for investigating the behavior of the very hot and dense nuclear matter formed in such collisions, in agreement with the collision geometry and collision energy; support for old and new comparisons of the predictions of the simulation codes and experimental results; and the insertion of new methods and ideas for these codes.
The major experiments performed at Relativistic Heavy Ion Collider (RHIC) from BNL, USA, as well as at the Large Hadron Collider (LHC) at CERN Geneva, have revealed the formation of quark–gluon plasma and a significant hydrodynamic collective behavior, similar to a nearly perfect fluid. Some clarifications regarding the evolution of the Universe have been obtained. In agreement with cosmological scenarios, including Gamow's “Big Bang" scenario, quark–gluon plasma appears a few microseconds after, and, therefore, it is possible to reproduce the conditions existing at that time in Universe in the laboratory.
There are other interesting nuclear matter phases, and the types of possible phase transitions are especially important for the study of the structure of nuclear matter and for the possible interactions. The current phase diagram of the nuclear matter still has areas that are not fully covered by the experimental data and results. Therefore, some laboratories have made efforts to achieve a scan over a wide range of energies, using the same experimental arrangement, with normalization in proton–proton collisions. There are remarkable results from RHIC-BNL, and from the efforts at SPS-CERN and the Tevatron (USA). There are also experimental results at lower energies from JINR Dubna, LNBL, etc.
On the other hand, the scientific community has proposed building new different acceleration and detector systems to cover other energies, especially in the phase transition region. Two projects are of great interest, namely FAIR (the Facility for Antiproton and Ion Research) at GSI Darmstadt (Germany) (www.gsi.de) and NICA (the Nuclotron-based Ion Collider Facility) at JINR Dubna (Russia). Both projects are now in an advanced stage, and the proposed experiments are already reaching the completion of the technical design reports for the detectors.
Therefore, the goal of the Editorial Board of Particles is stimulating and demanding, and authors should offer insights on these interesting aspects both experimentally and theoretically.
Prof. Dr. Alexandru Jipa
Guest Editor
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Keywords
- experiments
- collision geometry and bulk properties
- collision dynamics
- hydrodynamics
- thermalization
- flow
- correlations
- fluctuations
- phase transitions
- experimental signals
<p>Conventional Columbia plot. The strength of chiral symmetry breaking coincides with the axial symmetry breaking strength in the meson susceptibility functions with <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> because of the vanishing topological susceptibility. Thus, the simultaneous symmetry restoration between the chiral <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>U</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>A</mi> </msub> </mrow> </semantics></math> is realized on the <math display="inline"><semantics> <msub> <mi>m</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>d</mi> </mrow> </msub> </semantics></math> axis independently of the order of the chiral phase transition.</p> Full article ">Figure 2
<p>The temperature dependence of the susceptibility functions at the physical point (<math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>5.5</mn> </mrow> </semantics></math> MeV and <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>138</mn> </mrow> </semantics></math> MeV) for (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>/</mo> <msub> <mi>T</mi> <mi>pc</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>/</mo> <msub> <mi>T</mi> <mi>pc</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>. The pseusocritical temperature for the chiral crossover is observed to be <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>pc</mi> </msub> <mo>≃</mo> <mn>189</mn> </mrow> </semantics></math> MeV. The susceptibility functions are normalized by square of the pion decay constant (<math display="inline"><semantics> <mrow> <mo>≃</mo> <mn>93</mn> </mrow> </semantics></math> MeV), and the temperature axis is also normalized by <math display="inline"><semantics> <msub> <mi>T</mi> <mi>pc</mi> </msub> </semantics></math>, so all quantities are dimensionless to reduce the systematic uncertainty (approximately about 30%) associated with the present NJL model description of QCD. See also the text.</p> Full article ">Figure 3
<p>The plot showing the nontrivial coincidence between the chiral and axial indicators (of two types) in the crossover domain with the massless strange quark (<math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>5.5</mn> </mrow> </semantics></math> MeV and <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>pc</mi> </msub> <mo>≃</mo> <mn>144</mn> </mrow> </semantics></math> MeV). The topological susceptibility is exactly zero for all temperatures because the flavor-singlet nature associated with the massless strange quark. Scaling factors are applied on both horizontal and vertical axes in the same way as in <a href="#particles-07-00014-f002" class="html-fig">Figure 2</a>.</p> Full article ">Figure 4
<p>The plot showing the finiteness of the topological susceptibility along with the temperature dependence of the chiral and axial indicators in the crossover domain with a small strange quark mass (<math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>5.5</mn> </mrow> </semantics></math> MeV and <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msub> <mi>m</mi> <mi>l</mi> </msub> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>pc</mi> </msub> <mo>≃</mo> <mn>144</mn> </mrow> </semantics></math> MeV). The same scaling for two axes has been made as in <a href="#particles-07-00014-f002" class="html-fig">Figure 2</a>.</p> Full article ">Figure 5
<p>The plots clarifying the significant interference of the topological susceptibility to make the sizable discrepancy between the chiral and axial indicators in the crossover domain with the large strange quark mass (<math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>5.5</mn> </mrow> </semantics></math> MeV and <math display="inline"><semantics> <msub> <mi>m</mi> <mi>s</mi> </msub> </semantics></math> = 10 <math display="inline"><semantics> <msub> <mi>m</mi> <mi>l</mi> </msub> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>pc</mi> </msub> <mo>≃</mo> <mn>174</mn> </mrow> </semantics></math> MeV) for (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>/</mo> <msub> <mi>T</mi> <mi>pc</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>/</mo> <msub> <mi>T</mi> <mi>pc</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>. The manner of scaling axes is the same as in <a href="#particles-07-00014-f002" class="html-fig">Figure 2</a>.</p> Full article ">Figure 6
<p>The plots clarifying the trend of the nontrivial simultaneous restoration for the chiral and axial symmetries even in the chiral first-order phase-transition domain with <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> MeV and <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Panel (<b>a</b>) shows a jump in the mesons susceptibility functions at around <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi mathvariant="normal">c</mi> </msub> <mo>≃</mo> <mn>119</mn> </mrow> </semantics></math> MeV as a consequence of the first-order phase transition. The panel (<b>b</b>) closes up the temperature dependence for the chiral and axial indicators after the chiral phase transition. The manner of scaling axes is the same as in <a href="#particles-07-00014-f002" class="html-fig">Figure 2</a>. The trend induced by the interference of <math display="inline"><semantics> <msub> <mi>χ</mi> <mi>top</mi> </msub> </semantics></math> is similar to the one observed in the crossover domain in <a href="#particles-07-00014-f003" class="html-fig">Figure 3</a>.</p> Full article ">Figure 7
<p>The plots showing the still almost coincidence of the chiral and axial indicators for all temperature ranges, even in the first-order phase-transition domain with <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> MeV and <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msub> <mi>m</mi> <mi>l</mi> </msub> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi mathvariant="normal">c</mi> </msub> <mo>≃</mo> <mn>119</mn> </mrow> </semantics></math> MeV. The two displayed axes are scaled in the same way as explained in the caption of <a href="#particles-07-00014-f002" class="html-fig">Figure 2</a>. The trend induced by the interference of <math display="inline"><semantics> <msub> <mi>χ</mi> <mi>top</mi> </msub> </semantics></math> is similar to the one observed in the crossover domain in <a href="#particles-07-00014-f004" class="html-fig">Figure 4</a>.</p> Full article ">Figure 8
<p>The plots clarifying the significant interference of <math display="inline"><semantics> <msub> <mi>χ</mi> <mi>top</mi> </msub> </semantics></math> with the chiral and axial indicators in the first-order phase-transition domain with <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> MeV and <math display="inline"><semantics> <msub> <mi>m</mi> <mi>s</mi> </msub> </semantics></math> = 10 <math display="inline"><semantics> <msub> <mi>m</mi> <mi>l</mi> </msub> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi mathvariant="normal">c</mi> </msub> <mo>≃</mo> <mn>126</mn> </mrow> </semantics></math> MeV. The two displayed axes are scaled in the same way as explained in the caption of <a href="#particles-07-00014-f002" class="html-fig">Figure 2</a>. The trend induced by the interference of <math display="inline"><semantics> <msub> <mi>χ</mi> <mi>top</mi> </msub> </semantics></math> is similar to the one observed in the crossover domain in <a href="#particles-07-00014-f005" class="html-fig">Figure 5</a>.</p> Full article ">Figure 9
<p>The strange quark mass dependence on the difference between the axial indicator <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>χ</mi> <mrow> <mi>π</mi> <mo>−</mo> <mi>δ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>χ</mi> <mrow> <mi>η</mi> <mo>−</mo> <mi>σ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math>) and the chiral indicator <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>χ</mi> <mrow> <mi>η</mi> <mo>−</mo> <mi>δ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>χ</mi> <mrow> <mi>π</mi> <mo>−</mo> <mi>σ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math>) in (<b>a</b>) the crossover domain and (<b>b</b>) the first-order phase-transition domain. In the crossover domain with <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>/</mo> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0</mn> <mspace width="0.166667em"/> <mrow> <mo>(</mo> <mn>50</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, the pseudocritical temperature is evaluated as <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>pc</mi> </msub> <mo>≃</mo> <mn>144</mn> <mspace width="0.166667em"/> <mrow> <mo>(</mo> <mn>200</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> MeV, and the temperatures <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>300</mn> <mo>−</mo> <mn>400</mn> </mrow> </semantics></math> MeV displayed as in panel (<b>a</b>) correspond to <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>≃</mo> <mrow> <mo>(</mo> <mn>1.5</mn> <mo>−</mo> <mn>2.8</mn> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <msub> <mi>T</mi> <mi>pc</mi> </msub> </mrow> </semantics></math>. On the other hand, in the first-order phase-transition domain, the temperatures <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>240</mn> <mo>−</mo> <mn>300</mn> </mrow> </semantics></math> MeV as fixed in panel (<b>b</b>) correspond to <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>≃</mo> <mrow> <mo>(</mo> <mn>1.8</mn> <mo>−</mo> <mn>2.5</mn> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <msub> <mi>T</mi> <mi mathvariant="normal">c</mi> </msub> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi mathvariant="normal">c</mi> </msub> <mo>≃</mo> <mn>119</mn> <mspace width="0.166667em"/> <mrow> <mo>(</mo> <mn>130</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> MeV for <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>/</mo> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0</mn> <mspace width="0.166667em"/> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>The predicted chiral–axial phase diagram on the <math display="inline"><semantics> <msub> <mi>m</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>d</mi> </mrow> </msub> </semantics></math>-<math display="inline"><semantics> <msub> <mi>m</mi> <mi>s</mi> </msub> </semantics></math> plane, in which the discrepancy of the chiral and axial symmetry restorations at around <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>∼</mo> <mrow> <mo>(</mo> <mn>1.5</mn> <mo>−</mo> <mn>2.0</mn> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <msub> <mi>T</mi> <mrow> <mo>(</mo> <mi mathvariant="normal">p</mi> <mo>)</mo> <mi mathvariant="normal">c</mi> </mrow> </msub> </mrow> </semantics></math> is drawn by the shaded area. When the strength of the axial symmetry breaking deviates from the chiral breaking strength to be large, the shaded areas become thick. The nontrivial coincidence, as in Equation (<a href="#FD22-particles-07-00014" class="html-disp-formula">22</a>), is associated with the vanishing <math display="inline"><semantics> <msub> <mi>χ</mi> <mi>top</mi> </msub> </semantics></math>, which is located on the <math display="inline"><semantics> <msub> <mi>m</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>d</mi> </mrow> </msub> </semantics></math> axis. When the strange quark mass obtains a finite mass, the axial restoration deviates from the chiral restoration. At around <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>=</mo> <mi>O</mi> <mrow> <mo>(</mo> <mn>10</mn> <msub> <mi>m</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>, the axial restoration is much later than the chiral restoration because of the significant interference of <math display="inline"><semantics> <msub> <mi>χ</mi> <mi>top</mi> </msub> </semantics></math>. Namely, at the physical quark masses, the topological susceptibility provides the large discrepancy between the chiral and axial restorations in the meson susceptibilities.</p> Full article ">Figure 11
<p>The split in the restorations of the chiral <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>U</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> symmetry and the <math display="inline"><semantics> <mrow> <mi>U</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> axial symmetry at hot QCD. (<b>a</b>): Ordinary way to address the symmetry restorations. The ambiguous origin of the effective <math display="inline"><semantics> <mrow> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>A</mi> </msub> </mrow> </semantics></math> restoration is often measured by the topological susceptibility (which is normalized by the quark mass, <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>χ</mi> <mo stretchy="false">¯</mo> </mover> <mi>top</mi> </msub> <mo>=</mo> <mn>4</mn> <msub> <mi>χ</mi> <mi>top</mi> </msub> <mo>/</mo> <msubsup> <mi>m</mi> <mi>l</mi> <mn>2</mn> </msubsup> </mrow> </semantics></math>). (<b>b</b>): New point of view for symmetry restorations at <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Because of the anomalous Ward–Takahashi identity at hot QCD, the chiral <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>U</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> symmetry breaking exactly coincides with the <math display="inline"><semantics> <mrow> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>A</mi> </msub> </mrow> </semantics></math> symmetry breaking, and this coincidence holds for any temperatures. As a robust consequence, when the chiral <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>U</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> symmetry is restored at the (pseudo)critical temperature, the <math display="inline"><semantics> <mrow> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>A</mi> </msub> </mrow> </semantics></math> symmetry is simultaneously restored. Therefore, the limit <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> manifests the symmetry restorations on the quark mass plane: it can be unambiguously understood that the strange quark mass handles the split in the restorations of chiral <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>U</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> symmetry and the <math display="inline"><semantics> <mrow> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>A</mi> </msub> </mrow> </semantics></math> symmetry at hot QCD with the three quark flavors having finite masses.</p> Full article ">
<p>(Color online) Geometry of the pair production.</p> Full article ">Figure 2
<p>(Color online) An illustrative example when <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>R</mi> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. In the bottom panel we plot <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> <mi>f</mi> </mrow> </msub> </semantics></math> vs. <span class="html-italic">x</span> and the corresponding potential with full line (<math display="inline"><semantics> <mrow> <mo>±</mo> <msub> <mi>m</mi> <mi>T</mi> </msub> <mo>=</mo> <msub> <mi>m</mi> <mi>e</mi> </msub> </mrow> </semantics></math>—top panel, dashed and dotted lines) seen by the positron. The calculations are performed for <sup>12</sup>C+<sup>12</sup>C collisions.</p> Full article ">Figure 3
<p>(Color online) Tunneling probability for the positron as a function of the relative distance of the two C ions and <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mi>c</mi> <mo>.</mo> <mi>m</mi> <mo>.</mo> </mrow> </msub> </semantics></math>= 9.4 MeV and different values of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mi>E</mi> <mi>k</mi> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>(Color online) Upper limit for the integrated cross-section for <math display="inline"><semantics> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> </mrow> </semantics></math> production in <sup>12</sup>C+<sup>12</sup>C scattering below the Coulomb barrier for different values of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mi>E</mi> <mi>k</mi> </msub> </mrow> </semantics></math>. We stress that <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mi>E</mi> <mi>k</mi> </msub> <mo>≥</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">
<p>(Color online) The entropy <math display="inline"> <semantics> <mrow> <mi>H</mi> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>E</mi> <mo>|</mo> <mi>N</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics> </math> of Unruh radiation given by Equation (<a href="#FD17-particles-05-00014" class="html-disp-formula">17</a>) for fermions <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> as function of <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>/</mo> <mi>T</mi> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>M</mi> <mo>/</mo> <mi>T</mi> </mrow> </semantics> </math>.</p> Full article ">Figure 2
<p>(Color online) The same as <a href="#particles-05-00014-f001" class="html-fig">Figure 1</a> but for bosons. The spectrum of bosons contains (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics> </math> and (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics> </math> particles.</p> Full article ">Figure 3
<p>(Color online) Asymptotic behavior of entropy <math display="inline"> <semantics> <mrow> <mi>H</mi> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>E</mi> <mo>|</mo> <mi>N</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics> </math> given by Equation (<a href="#FD23-particles-05-00014" class="html-disp-formula">23</a>) at <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>→</mo> <mn>0</mn> </mrow> </semantics> </math> as function of <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>/</mo> <mi>T</mi> </mrow> </semantics> </math>.</p> Full article ">Figure 4
<p>(Color online) High-temperature asymptotics of the entropy <math display="inline"> <semantics> <mrow> <mi>H</mi> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>E</mi> <mo>|</mo> <mi>N</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics> </math> of Unruh radiation given by Equation (<a href="#FD28-particles-05-00014" class="html-disp-formula">28</a>) for fermions <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> as a function of <span class="html-italic">m</span> and <span class="html-italic">M</span>.</p> Full article ">Figure 5
<p>(Color online) The same as <a href="#particles-05-00014-f004" class="html-fig">Figure 4</a> but for bosons with (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics> </math> and (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics> </math> particles in the spectrum.</p> Full article ">
<p>(Color online) The transverse momentum distribution of charged particles produced in Au + Au collisions for energies ranging from <math display="inline"><semantics> <mrow> <msqrt> <msub> <mi>s</mi> <mrow> <mi>N</mi> <mi>N</mi> </mrow> </msub> </msqrt> <mo>=</mo> <mn>7.7</mn> </mrow> </semantics></math> to 200 GeV and Cu + Cu collisions at <math display="inline"><semantics> <mrow> <msqrt> <msub> <mi>s</mi> <mrow> <mi>N</mi> <mi>N</mi> </mrow> </msub> </msqrt> <mo>=</mo> <mn>62.4</mn> </mrow> </semantics></math> and 200 GeV at the RHIC. Scale factors are applied for better visibility. The curves are the fit results from Equation (<a href="#FD4-particles-05-00013" class="html-disp-formula">4</a>). The experimental data are taken from Refs. [<a href="#B1-particles-05-00013" class="html-bibr">1</a>,<a href="#B2-particles-05-00013" class="html-bibr">2</a>,<a href="#B3-particles-05-00013" class="html-bibr">3</a>,<a href="#B4-particles-05-00013" class="html-bibr">4</a>]. The corresponding ratios of data/fit are also shown.</p> Full article ">Figure 2
<p>(Color online) Same as <a href="#particles-05-00013-f001" class="html-fig">Figure 1</a> but for Pb + Pb collisions at <math display="inline"><semantics> <mrow> <msqrt> <msub> <mi>s</mi> <mrow> <mi>N</mi> <mi>N</mi> </mrow> </msub> </msqrt> <mo>=</mo> <mn>2.76</mn> </mrow> </semantics></math> and 5.02 TeV and Xe + Xe collisions at <math display="inline"><semantics> <mrow> <msqrt> <msub> <mi>s</mi> <mrow> <mi>N</mi> <mi>N</mi> </mrow> </msub> </msqrt> <mo>=</mo> <mn>5.44</mn> </mrow> </semantics></math> TeV at LHC. The experimental data are taken from Refs. [<a href="#B5-particles-05-00013" class="html-bibr">5</a>,<a href="#B6-particles-05-00013" class="html-bibr">6</a>].</p> Full article ">Figure 3
<p>(Color online) The collision energy dependence of the temperature <span class="html-italic">T</span> and nonextensive parameter <span class="html-italic">q</span> for the collision systems in <a href="#particles-05-00013-f001" class="html-fig">Figure 1</a> and <a href="#particles-05-00013-f002" class="html-fig">Figure 2</a> at the most central collisions. See text for the lines.</p> Full article ">Figure 4
<p>(Color online) The centrality (0 represents the most central collisions) dependence of the <span class="html-italic">T</span> and <span class="html-italic">q</span> in Au + Au collisions at <math display="inline"><semantics> <mrow> <msqrt> <msub> <mi>s</mi> <mrow> <mi>N</mi> <mi>N</mi> </mrow> </msub> </msqrt> <mo>=</mo> <mn>7.7</mn> <mo>−</mo> <mn>200</mn> </mrow> </semantics></math> GeV, Cu + Cu collisions at <math display="inline"><semantics> <mrow> <msqrt> <msub> <mi>s</mi> <mrow> <mi>N</mi> <mi>N</mi> </mrow> </msub> </msqrt> <mo>=</mo> <mn>62.4</mn> </mrow> </semantics></math>, 200 GeV, Pb + Pb collisions <math display="inline"><semantics> <mrow> <msqrt> <msub> <mi>s</mi> <mrow> <mi>N</mi> <mi>N</mi> </mrow> </msub> </msqrt> <mo>=</mo> <mn>2.76</mn> </mrow> </semantics></math>, 5.02 TeV and Xe + Xe collisions at <math display="inline"><semantics> <mrow> <msqrt> <msub> <mi>s</mi> <mrow> <mi>N</mi> <mi>N</mi> </mrow> </msub> </msqrt> <mo>=</mo> <mn>5.44</mn> </mrow> </semantics></math> TeV. The curves in (<b>a</b>,<b>b</b>) are the parabolic fits and the lines in (<b>c</b>,<b>d</b>) are the linear fits.</p> Full article ">Figure 5
<p>(Color online) Nonextensive parameter <span class="html-italic">q</span> dependence of the temperature divided by the natural logarithm of collision energy <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>/</mo> <mi>l</mi> <mi>n</mi> <msqrt> <mi>s</mi> </msqrt> </mrow> </semantics></math> in A + A collisions with different centrality. The curve is the fit results and the fit function is indicated in the legend.</p> Full article ">
<p>(<b>a</b>) Layout of the double-focusing magnetic spectrometer KaoS with its detector system: time-of-flight (TOF) start and stop detectors, 3 multi-wire proportional chambers (MWPC), Cherenkov detectors with water, lucite and silica aerogel radiators, two hodoscopes for event characterization at target (for large angles) and 7 m downstream the target (for small angles, not shown) [<a href="#B32-particles-05-00003" class="html-bibr">32</a>]. (<b>b</b>) Photo of the setup.</p> Full article ">Figure 2
<p>Examples of the acceptance of the kaon spectrometer in the plane transverse momentum versus normalized rapidity y/y<sub>beam</sub> for several laboratory angles θ<sub>lab</sub> (as indicated) and for various magnetic field settings at 1.0A GeV, at 1.5A GeV and at 1.93A GeV beam energy. The angular acceptance is ∆θ<sub>lab</sub> = ±4°, corresponding to the width of the pink bands.</p> Full article ">Figure 3
<p>Pion multiplicity per participating nucleon measured in nucleus–nucleus collisions (symbols) and in nucleon–nucleon collisions as a function of available energy in the NN system (taken from [<a href="#B33-particles-05-00003" class="html-bibr">33</a>]).</p> Full article ">Figure 4
<p>Differential pion production cross-sections measured at various polar emission angles in Au+Au collisions at 1.5A GeV (π<sup>+</sup> left panel and π<sup>−</sup> center panel) and in C+C collisions at 1A GeV (π<sup>+</sup> right panel) as a function of the kinetic energy in the c.m. system (black triangles) in comparison to results of URQMD transport calculations (green triangles) [<a href="#B34-particles-05-00003" class="html-bibr">34</a>].</p> Full article ">Figure 5
<p>(<b>a</b>) Three snapshots of a Au+Au collision with a beam kinetic energy of 1A GeV (impact parameter b = 7 fm) calculated with the QMD transport code for 4 fm/c (left), 10 fm/c (center) and 16 fm/c (right). The pions are emitted in the reaction plane at backward angles corresponding to a particular detector position. (<b>b</b>) Pion number ratio N<sup>π</sup><sub>proj</sub>/N<sup>π</sup>t<sub>arg</sub> measured as function of transverse momentum in peripheral Au+Au collisions (b ≥ 5.7 fm) at 1A GeV at target rapidity. N<sup>π</sup><sub>proj</sub> and N<sup>π</sup>t<sub>arg</sub> denote the numbers of pions emitted to the projectile and to the target side, respectively, within a cone of ±45° [<a href="#B35-particles-05-00003" class="html-bibr">35</a>].</p> Full article ">Figure 6
<p>(<b>a</b>) Azimuthal distributions of positively charged pions for peripheral, semi-central and central collisions (from top to bottom) measured in Au+Au collisions at 1A GeV [<a href="#B37-particles-05-00003" class="html-bibr">37</a>]. The ordinate is linear starting at zero. Left column: π<sup>+</sup> in the range 160 < p<sub>T</sub> < 260 MeV/c. Right column: π<sup>+</sup> in the range 260 < p<sub>T</sub> < 600 MeV/c. Solid lines: fits to the data with cos (φ) and cos (2φ) terms. φ = 0° and φ = ±180° represent emission of pions parallel to the reaction plane and φ = ±90° corresponds to emission of pions perpendicular to the reaction plane. (<b>b</b>) Illustration of the particle emission pattern for semi-central collisions at intermediate beam energies perpendicular to the reaction plane (“off-plane squeeze-out”) and parallel to the reaction plane (“bounce off”).</p> Full article ">Figure 7
<p>K<sup>+</sup> and K<sup>−</sup> multiplicity per number of participating nucleons as a function of the available energy above threshold in first-chance collisions for C+C and Ni+Ni collisions (for symbols, see legend) and parameterizations of the kaon production cross sections in nucleon–nucleon collisions (for lines, see insert). Taken from [<a href="#B40-particles-05-00003" class="html-bibr">40</a>].</p> Full article ">Figure 8
<p>Inclusive invariant cross-sections at mid-rapidity as a function of the kinetic energy E<sub>c</sub>.<sub>m</sub>. − m<sub>0</sub>c<sup>2</sup> for K<sup>+</sup> mesons (<b>a</b>) and for K<sup>−</sup> mesons (<b>b</b>) for the various collision systems and beam energies measured. Mid-rapidity data were selected by the condition θ<sub>c</sub>.<sub>m</sub>.= 90° ± 10° from measurements at different polar angles [<a href="#B41-particles-05-00003" class="html-bibr">41</a>]. The lines represent fits to the data (see text).</p> Full article ">Figure 9
<p>Multiplicities of K<sup>+</sup> (full symbols) and of K<sup>−</sup> mesons (open symbols) per mass number A of the collision system as a function of the beam energy. The lines represent fits to the data [<a href="#B41-particles-05-00003" class="html-bibr">41</a>].</p> Full article ">Figure 10
<p>Azimuthal distribution of K<sup>+</sup> mesons measured in semi-central Au+Au collisions at 1A GeV (full dots). The kaons are analyzed for transverse momenta within a range of 0.2 GeV/c ≤ p<sub>t</sub> ≤ 0.8 GeV/c and for the normalized rapidity ranges of 0.4 ≤ y/y<sub>proj</sub> ≤ 0.6 (<b>a</b>) and 0.2 ≤ y/y<sub>proj</sub> ≤ 0.8 (<b>b</b>) [<a href="#B7-particles-05-00003" class="html-bibr">7</a>]. The lines show the results of transport calculations using a RBUU model (left; [<a href="#B42-particles-05-00003" class="html-bibr">42</a>]) and a QMD model (right; [<a href="#B43-particles-05-00003" class="html-bibr">43</a>]), which both take into account rescattering; QMD also calculates Coulomb effects. Solid and dashed lines: calculations with and without in-medium K<sup>+</sup>N potential, respectively. Taken from [<a href="#B40-particles-05-00003" class="html-bibr">40</a>].</p> Full article ">Figure 11
<p>Azimuthal angular distributions of π<sup>+</sup>, K<sup>+</sup> and K<sup>−</sup>mesons (from <b>left</b> to <b>right</b>) measured in semi-central Ni+Ni collisions at 1.93A·GeV [<a href="#B46-particles-05-00003" class="html-bibr">46</a>]. The mesons are measured within a rapidity range of 0.3 < y/y<sub>beam</sub> < 0.7 and a momentum range of 0.2 GeV/c < p<sub>t</sub> < 0.8 GeV/c. The data are fitted using the first two components of a Fourier series dN/dΦ∼2 v<sub>1</sub> cos (φ) + 2 v<sub>2</sub> cos (2φ). The resulting values for v<sub>1</sub> and v<sub>2</sub> are indicated.</p> Full article ">Figure 12
<p>Multiplicity density distributions of K<sup>+</sup> mesons (upper panel), K<sup>−</sup> mesons (center panel) and the K<sup>+</sup>/K<sup>−</sup> ratio (lower panel) for near-central (<span class="html-italic">b</span> < 4.4 fm) Ni+Ni collisions at 1.93A GeV, measured by KaoS (red circles) [<a href="#B47-particles-05-00003" class="html-bibr">47</a>] and FOPI (green squares) [<a href="#B48-particles-05-00003" class="html-bibr">48</a>,<a href="#B49-particles-05-00003" class="html-bibr">49</a>]. The measured data (full symbols) are mirrored at <span class="html-italic">y</span><sub>CM</sub> = 0 (open symbols). The data are compared to BUU transport calculations [<a href="#B50-particles-05-00003" class="html-bibr">50</a>]. Solid lines: with in-medium effects. Dotted lines: without in-medium effects. Taken from [<a href="#B33-particles-05-00003" class="html-bibr">33</a>].</p> Full article ">Figure 13
<p>Upper panel: Density in the reaction volume as function of time for a central Au+Au collision at 1A GeV as calculated by a RBUU transport model. Lower panel: Multiplicities of produced Δ resonances (green dotted line), pions (red dashed line) and K<sup>+</sup> mesons (blue line) as functions of time.</p> Full article ">Figure 14
<p>(<b>a</b>) Production cross-sections of K<sup>+</sup> mesons measured in Au+Au and C+C collisions as functions of the projectile energy per nucleon (black diamonds). The data are compared to QMD calculations with (full symbols) and without (open symbols) kaon in-medium modifications, assuming a soft EOS (K<sub>nm</sub> = 200 MeV; blue dots) or a hard EOS (K<sub>nm</sub> = 380 MeV; cyan squares). (<b>b</b>) Ratio of the K<sup>+</sup> multiplicity per mass number in Au+Au over C+C collisions as a function of beam energy. The data are compared to different QMD calculations assuming a soft EOS (K<sub>nm</sub> = 200 MeV; red symbols) or a hard EOS (K<sub>nm</sub> = 380 MeV; blue symbols). Taken from [<a href="#B57-particles-05-00003" class="html-bibr">57</a>].</p> Full article ">Figure 15
<p>Binding energy as a function of nuclear matter density in units of ρ<sub>0</sub>. The lines represent the results of various calculations for neutron matter (upper curves) and for symmetric matter (lower curves) [<a href="#B62-particles-05-00003" class="html-bibr">62</a>]. Lower green area: EOS for symmetric matter as extracted from data of the KaoS [<a href="#B54-particles-05-00003" class="html-bibr">54</a>,<a href="#B55-particles-05-00003" class="html-bibr">55</a>] and FOPI [<a href="#B59-particles-05-00003" class="html-bibr">59</a>] experiments. Upper green area: Symmetry energy E<sub>sym</sub> as extracted from the data of the ASY–EOS experiment [<a href="#B61-particles-05-00003" class="html-bibr">61</a>] added to the experimental EOS for symmetric matter (see text).</p> Full article ">Figure 16
<p>EOS of symmetric nuclear matter expressed as pressure versus baryon density. Grey hatched area: constraint from proton flow data taken at AGS [<a href="#B11-particles-05-00003" class="html-bibr">11</a>,<a href="#B12-particles-05-00003" class="html-bibr">12</a>]. Yellow area: Constraint from fragment flow and kaon data taken at GSI [<a href="#B54-particles-05-00003" class="html-bibr">54</a>,<a href="#B55-particles-05-00003" class="html-bibr">55</a>,<a href="#B59-particles-05-00003" class="html-bibr">59</a>]. Red line: Hard EOS. Blue line: Soft EOS [<a href="#B12-particles-05-00003" class="html-bibr">12</a>].</p> Full article ">
<p>Near- and away-side peak yield (<b>top row</b>), width (<b>middle row</b>), <math display="inline"><semantics> <mi>β</mi> </semantics></math> parameter and baseline (<b>bottom row</b>) of D–h correlations from simulations with different parton level contributions in pp collisions at <math display="inline"><semantics> <mrow> <msqrt> <mi>s</mi> </msqrt> <mo>=</mo> <mn>5.02</mn> </mrow> </semantics></math> TeV as a function of the D meson <math display="inline"><semantics> <msub> <mi>p</mi> <mi mathvariant="normal">T</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mn>1</mn> <mo><</mo> <msubsup> <mi>p</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mrow> <mi>assoc</mi> <mo>.</mo> </mrow> </msubsup> <mo><</mo> <mn>2</mn> </mrow> </semantics></math> GeV/<span class="html-italic">c</span>.</p> Full article ">Figure 2
<p>Associated yield of D–h correlations from simulations with different parton level contributions of prompt D mesons with hadrons in simulated pp collisions at <math display="inline"><semantics> <mrow> <msqrt> <mi>s</mi> </msqrt> <mo>=</mo> <mn>5.02</mn> </mrow> </semantics></math> TeV for <math display="inline"><semantics> <mrow> <mn>1</mn> <mo><</mo> <msubsup> <mi>p</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mrow> <mi>assoc</mi> <mo>.</mo> </mrow> </msubsup> <mo><</mo> <mn>2</mn> </mrow> </semantics></math> GeV/<span class="html-italic">c</span> and <math display="inline"><semantics> <mrow> <mn>3</mn> <mo><</mo> <msubsup> <mi>p</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">D</mi> </msubsup> <mo><</mo> <mn>5</mn> </mrow> </semantics></math> GeV/<span class="html-italic">c</span>.</p> Full article ">Figure 3
<p>Near-side peak yield of simulated D–h correlations with different heavy-flavour fragmentation models in pp collision at <math display="inline"><semantics> <mrow> <msqrt> <mi>s</mi> </msqrt> <mo>=</mo> <mn>5.02</mn> </mrow> </semantics></math> TeV, as a function of the D-meson <math display="inline"><semantics> <msub> <mi>p</mi> <mi mathvariant="normal">T</mi> </msub> </semantics></math>, for <math display="inline"><semantics> <mrow> <msubsup> <mi>p</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mrow> <mi>assoc</mi> <mo>.</mo> </mrow> </msubsup> <mo>></mo> <mn>0.3</mn> </mrow> </semantics></math> GeV/<span class="html-italic">c</span>.</p> Full article ">Figure 4
<p>Near-side peak yield of simulated D–h correlations in pp collision at <math display="inline"><semantics> <mrow> <msqrt> <mi>s</mi> </msqrt> <mo>=</mo> <mn>5.02</mn> </mrow> </semantics></math> TeV, as a function of the D-meson <math display="inline"><semantics> <msub> <mi>p</mi> <mi mathvariant="normal">T</mi> </msub> </semantics></math>, for <math display="inline"><semantics> <mrow> <msubsup> <mi>p</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mrow> <mi>assoc</mi> <mo>.</mo> </mrow> </msubsup> <mo>></mo> <mn>0.3</mn> </mrow> </semantics></math> GeV/<span class="html-italic">c</span>.</p> Full article ">Figure 5
<p>Near- and away-side peak yield (<b>top row</b>), width (<b>middle row</b>), <math display="inline"><semantics> <mi>β</mi> </semantics></math> parameter and baseline (<b>bottom row</b>) of prompt and non-prompt D meson correlations with hadrons in simulated pp collisions at <math display="inline"><semantics> <mrow> <msqrt> <mi>s</mi> </msqrt> <mo>=</mo> <mn>5.02</mn> </mrow> </semantics></math> TeV as a function of the D meson <math display="inline"><semantics> <msub> <mi>p</mi> <mi mathvariant="normal">T</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <msubsup> <mi>p</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mrow> <mi>assoc</mi> <mo>.</mo> </mrow> </msubsup> <mo>></mo> <mn>0.3</mn> </mrow> </semantics></math> GeV/<span class="html-italic">c</span>.</p> Full article ">Figure 6
<p>Associated yield of prompt and non-prompt D meson correlations with hadrons in simulated pp collisions at <math display="inline"><semantics> <mrow> <msqrt> <mi>s</mi> </msqrt> <mo>=</mo> <mn>5.02</mn> </mrow> </semantics></math> TeV for <math display="inline"><semantics> <mrow> <msubsup> <mi>p</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mrow> <mi>assoc</mi> <mo>.</mo> </mrow> </msubsup> <mo>></mo> <mn>0.3</mn> </mrow> </semantics></math> GeV/<span class="html-italic">c</span> and <math display="inline"><semantics> <mrow> <mn>3</mn> <mo><</mo> <msubsup> <mi>p</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">D</mi> </msubsup> <mo><</mo> <mn>5</mn> </mrow> </semantics></math> GeV/<span class="html-italic">c</span>.</p> Full article ">
<p>Three regimes of the energy dependence of the diffraction cone slope parameter <math display="inline"><semantics> <mrow> <mi>B</mi> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </semantics></math> in the Regge-pole model amended with contribution of multi–Pomeron exchanges in the LHC energy range. <span class="html-italic">Quantitative justification for the second break can be found in</span> [<a href="#B1-particles-04-00033" class="html-bibr">1</a>,<a href="#B2-particles-04-00033" class="html-bibr">2</a>,<a href="#B3-particles-04-00033" class="html-bibr">3</a>].</p> Full article ">
<p>The lowest-order correction to the vacuum polarization <math display="inline"><semantics> <msup> <mi mathvariant="sans-serif">Π</mi> <mrow> <mi>ρ</mi> <mi>σ</mi> </mrow> </msup> </semantics></math> of photons, due to quarks of the flavor <span class="html-italic">q</span>.</p> Full article ">