Towards a Warm Holographic Equation of State by an Einstein–Maxwell-Dilaton Model
<p>Comparison of the EMd model results with lattice data [<a href="#B24-symmetry-16-00999" class="html-bibr">24</a>] (crosses) for <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>/</mo> <mi>T</mi> <mo>=</mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mi>top</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>6</mn> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mi>bottom</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>/</mo> <msup> <mi>T</mi> <mn>4</mn> </msup> </mrow> </semantics></math> (left column), <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>B</mi> </msub> <mo>/</mo> <msup> <mi>T</mi> <mn>3</mn> </msup> </mrow> </semantics></math> (middle column), and <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>/</mo> <mi>p</mi> </mrow> </semantics></math> (right column) as a function of <span class="html-italic">T</span>. Note the different scales for <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>B</mi> </msub> <mo>/</mo> <msup> <mi>T</mi> <mn>3</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>/</mo> <mi>p</mi> </mrow> </semantics></math>.</p> "> Figure 1 Cont.
<p>Comparison of the EMd model results with lattice data [<a href="#B24-symmetry-16-00999" class="html-bibr">24</a>] (crosses) for <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>/</mo> <mi>T</mi> <mo>=</mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mi>top</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>6</mn> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mi>bottom</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>/</mo> <msup> <mi>T</mi> <mn>4</mn> </msup> </mrow> </semantics></math> (left column), <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>B</mi> </msub> <mo>/</mo> <msup> <mi>T</mi> <mn>3</mn> </msup> </mrow> </semantics></math> (middle column), and <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>/</mo> <mi>p</mi> </mrow> </semantics></math> (right column) as a function of <span class="html-italic">T</span>. Note the different scales for <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>B</mi> </msub> <mo>/</mo> <msup> <mi>T</mi> <mn>3</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>/</mo> <mi>p</mi> </mrow> </semantics></math>.</p> "> Figure 2
<p>Contour plots of scaled entropy density <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>/</mo> <msup> <mi>T</mi> <mn>3</mn> </msup> </mrow> </semantics></math> (<b>left top panel</b>), baryon density <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>B</mi> </msub> <mo>/</mo> <msup> <mi>T</mi> <mn>3</mn> </msup> </mrow> </semantics></math> (<b>right top panel</b>), entropy per baryon <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>/</mo> <msub> <mi>n</mi> <mi>B</mi> </msub> </mrow> </semantics></math> (<b>left bottom panel</b>, relevant for adiabatic expansion), and pressure <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>/</mo> <msup> <mi>T</mi> <mn>4</mn> </msup> </mrow> </semantics></math> (<b>right bottom panel</b>) over the <span class="html-italic">T</span>-<math display="inline"><semantics> <msub> <mi>μ</mi> <mi>B</mi> </msub> </semantics></math> plane. The CEP is depicted as a bullet and the solid black curve is the emerging FOPT. The labeling numbers “<span class="html-italic">N</span>” mean <math display="inline"><semantics> <msup> <mn>10</mn> <mi>N</mi> </msup> </semantics></math> of the respective quantity. Note the weak dependence of <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>/</mo> <msup> <mi>T</mi> <mn>3</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>/</mo> <msup> <mi>T</mi> <mn>4</mn> </msup> </mrow> </semantics></math> on <math display="inline"><semantics> <msub> <mi>μ</mi> <mi>B</mi> </msub> </semantics></math> to the left of the FOPT at <math display="inline"><semantics> <mrow> <mi>T</mi> <mspace width="3.33333pt"/> <mo><</mo> <mspace width="3.33333pt"/> <mn>100</mn> </mrow> </semantics></math> MeV. The crosses depict results of the lattice QCD calculations [<a href="#B24-symmetry-16-00999" class="html-bibr">24</a>]. The scaled energy density, <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>/</mo> <msup> <mi>T</mi> <mn>4</mn> </msup> <mo>=</mo> <mo>−</mo> <mi>p</mi> <mo>/</mo> <msup> <mi>T</mi> <mn>4</mn> </msup> <mo>+</mo> <mi>s</mi> <mo>/</mo> <msup> <mi>T</mi> <mn>3</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>/</mo> <mi>T</mi> <mo>)</mo> </mrow> <msub> <mi>n</mi> <mi>B</mi> </msub> <mo>/</mo> <msup> <mi>T</mi> <mn>3</mn> </msup> </mrow> </semantics></math>, can be inferred from the displayed information.</p> "> Figure 3
<p>Contour plot of the EoS as isobars <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>)</mo> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> </mrow> </semantics></math> (<b>left panel</b>) and iso-energy density curves <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>)</mo> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> </mrow> </semantics></math> (<b>right panel</b>) over the <span class="html-italic">T</span>-<math display="inline"><semantics> <msub> <mi>μ</mi> <mi>B</mi> </msub> </semantics></math> plane. The CEP, FOPT, line style, and meaning of labeling (here in units of MeV/fm<sup>3</sup>) are as in <a href="#symmetry-16-00999-f002" class="html-fig">Figure 2</a>. Note again the weak dependence on <math display="inline"><semantics> <msub> <mi>μ</mi> <mi>B</mi> </msub> </semantics></math> to the left of the FOPT at <math display="inline"><semantics> <mrow> <mi>T</mi> <mo><</mo> <mn>100</mn> </mrow> </semantics></math> MeV. The crosses depict results of the lattice QCD calculations [<a href="#B24-symmetry-16-00999" class="html-bibr">24</a>].</p> "> Figure 4
<p>(<b>Left panel</b>): Energy density <span class="html-italic">e</span> (solid curves) as a function of temperature <span class="html-italic">T</span> along the “safe” isobars <math display="inline"><semantics> <mrow> <mi>T</mi> <mrow> <mo>(</mo> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>)</mo> </mrow> <msub> <mrow> <mo>|</mo> </mrow> <mrow> <mi>p</mi> <mo>=</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> </mrow> </msub> </mrow> </semantics></math> (see <a href="#symmetry-16-00999-f003" class="html-fig">Figure 3</a>—left) for various values of <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>125</mn> </mrow> </semantics></math> (black), 150 (cyan), 175 (yellow), and 200 MeV (magenta), and, thus, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. In addition, the case of a “less reliable” isobar with <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> MeV is also displayed (red). The right-hand side endpoints (“o”) are for <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, both for <span class="html-italic">e</span> and pressure <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> </mrow> </semantics></math> (horizontal thin lines with the same color code as the corresponding energy density). The difference of <span class="html-italic">e</span> and <span class="html-italic">p</span> (both in units of MeV/fm<sup>3</sup>) in the employed log scale delivers directly <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>/</mo> <mi>p</mi> </mrow> </semantics></math> as a function of <span class="html-italic">T</span> along the respective isobar. Equally well, <span class="html-italic">e</span> and <span class="html-italic">p</span> for a selected constant value of <span class="html-italic">T</span> can be read off, thus providing the iso-thermal EoS <math display="inline"><semantics> <msub> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <mi>e</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mi>T</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, exhibited in the (<b>right panel</b>) for various temperatures as provided by labels. The right-hand side endpoints “+” are for <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>=</mo> <mn>2000</mn> </mrow> </semantics></math> MeV. One could also combine the results of <a href="#symmetry-16-00999-f002" class="html-fig">Figure 2</a> along cuts of <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> </mrow> </semantics></math> to arrive at the same picture. The crosses depict results of the lattice QCD calculations [<a href="#B24-symmetry-16-00999" class="html-bibr">24</a>] in both panels. The bullet depicts the onset point of the perturbative QCD regime for <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Nuclear many-body theory is expected to apply below the left bottom corner.</p> "> Figure A1
<p>Illustration of expected isobars <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>)</mo> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> </mrow> </semantics></math> over the <span class="html-italic">T</span>-<math display="inline"><semantics> <msub> <mi>μ</mi> <mi>B</mi> </msub> </semantics></math> plane in a toy model. The heavy solid bar on the <span class="html-italic">T</span> axis indicates the region, where reliable QCD input data (e.g., <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>(</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>) are at our disposal. The continuation to <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics></math> is controlled by lattice data in the hatched region (with sections of rays <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>/</mo> <mi>T</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> </mrow> </semantics></math> highlighted). Isobars not emerging from the heavy solid vertical bar or not running a noticeable section through the hatched control region are to be considered as less reliable (dashed or dotted curves). Irrespective of the EoS on the <span class="html-italic">T</span> axis, such a mapping by “laminar curves” <math display="inline"><semantics> <mrow> <mi>T</mi> <mrow> <mo>(</mo> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>)</mo> </mrow> <msub> <mrow> <mo>|</mo> </mrow> <mrow> <mi>p</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> </mrow> </msub> </mrow> </semantics></math> (solid curves) would allow us to arrive unambiguously at the cool EoS at <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, or any other cut through the <span class="html-italic">T</span>-<math display="inline"><semantics> <msub> <mi>μ</mi> <mi>B</mi> </msub> </semantics></math> plane, thus also providing a warm EoS for neutron star merger dynamics.</p> "> Figure A2
<p>The stable branches of scaled density <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>B</mi> </msub> <mo>/</mo> <msup> <mi>T</mi> <mn>3</mn> </msup> </mrow> </semantics></math> (left panel) and scaled pressure <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>/</mo> <msup> <mi>T</mi> <mn>4</mn> </msup> </mrow> </semantics></math> as a function of temperature <span class="html-italic">T</span> for various values of <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>B</mi> </msub> <mo>=</mo> <mi>n</mi> <mspace width="0.166667em"/> <mn>500</mn> </mrow> </semantics></math> MeV for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (blue), 1 (green), 2 (red), 3 (cyan), and 4 (magenta). The crosses depict the results of the lattice QCD calculations [<a href="#B24-symmetry-16-00999" class="html-bibr">24</a>].</p> "> Figure A3
<p>Contour plots of <math display="inline"><semantics> <msup> <mi>χ</mi> <mn>2</mn> </msup> </semantics></math> with respect to scaled entropy density (<b>left panel</b>), <math display="inline"><semantics> <msup> <mi>L</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (<b>middle panel</b>), and <math display="inline"><semantics> <msub> <mi>κ</mi> <mn>5</mn> </msub> </semantics></math> (<b>right panel</b>, in units of <math display="inline"><semantics> <msup> <mi>L</mi> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </semantics></math>) for the dilaton potential function Equation (<a href="#FD4-symmetry-16-00999" class="html-disp-formula">4</a>) with local maximum of <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">W</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>3.25</mn> </mrow> </semantics></math> as side conditions. The dashed line depicts the locus of <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> determined by <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi mathvariant="script">W</mi> <mi>m</mi> </msub> <mo form="prefix">exp</mo> <mrow> <mo>{</mo> <mi>γ</mi> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>}</mo> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>−</mo> <mi>γ</mi> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> </mrow> </semantics></math>, i.e., for <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics></math>, an unintended thermal phase transition is excluded since, beyond the maximum, <math display="inline"><semantics> <mrow> <mi mathvariant="script">W</mi> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </semantics></math> is smoothly and monotonously approaching zero at <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>. The bullet in the <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo><</mo> <mn>0</mn> </mrow> </semantics></math> region is for the parameter choice of <math display="inline"><semantics> <msub> <mi>p</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <mi>γ</mi> </semantics></math> listed below Equation (5), which facilitates <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">W</mi> <mi>m</mi> </msub> <mo>≈</mo> <mn>0.6</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>≈</mo> <mn>3.25</mn> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. Holographic Einstein–Maxwell-Dilaton Model
3. Numerical Results: EoS
3.1. CEP Location and FOPT
3.2. Scaled Entropy, Density, Pressure, and Specific Entropy
3.3. Isobars and Iso-Energy Lines
3.4. Warm EoS
4. Conclusions and Summary
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. A Toy Model of Isobars
Appendix B. Details of the EMd Model
Appendix C. Density and Pressure at FOPT
Appendix D. Various Dilaton Potential Parameterizations
References
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Zöllner, R.; Kämpfer, B. Towards a Warm Holographic Equation of State by an Einstein–Maxwell-Dilaton Model. Symmetry 2024, 16, 999. https://doi.org/10.3390/sym16080999
Zöllner R, Kämpfer B. Towards a Warm Holographic Equation of State by an Einstein–Maxwell-Dilaton Model. Symmetry. 2024; 16(8):999. https://doi.org/10.3390/sym16080999
Chicago/Turabian StyleZöllner, Rico, and Burkhard Kämpfer. 2024. "Towards a Warm Holographic Equation of State by an Einstein–Maxwell-Dilaton Model" Symmetry 16, no. 8: 999. https://doi.org/10.3390/sym16080999