# Conformal Anomaly in Yang-Mills Theory and Thermodynamics of Open Confining Strings

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## Abstract

**:**

## 1. Introduction

## 2. The Conformal Anomaly and Thermodynamics of Gluons

#### 2.1. Conformal Anomaly with Isotropic Renormalization Scale: Equation of State of Gluon Plasma

#### 2.2. Conformal Anomaly with Anisotropic Renormalization Scales: Energy and Pressure via Condensates

## 3. Gluon Energy and Pressure via Gluon Condensates in Presence of Confining String

#### 3.1. Thermodynamics from the Conformal Anomaly

#### 3.2. Generalization of Michael-Rothe Sum Rules

- (i)
- (ii)
- (iii)
- Finally, the pair Equations (31) and (32) describes the new finite-temperature sum rules.

#### 3.3. New Sum Rules and Exact Relations for Gluon Thermodynamics around Static Sources

#### 3.4. Negative Pressure Excess

#### 3.5. Multiquarks and a Single Quark

#### 3.6. A Few Phenomenological Examples

#### 3.6.1. Deconfinement Phase

#### 3.6.2. Confinement Phase at Finite Temperature

#### 3.6.3. Zero Temperature

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Gyulassy, M.; McLerran, L. New forms of QCD matter discovered at RHIC. Nucl. Phys. A
**2005**, 750, 30–63. [Google Scholar] [CrossRef] [Green Version] - Müller, B.; Schukraft, J.; Wyslouch, B. First results from Pb + Pb collisions at the LHC. Ann. Rev. Nucl. Part. Sci.
**2012**, 62, 361–396. [Google Scholar] [CrossRef] [Green Version] - Jacak, B.V.; Müller, B. The exploration of hot nuclear matter. Nature
**2012**, 337, 310–314. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Romatschke, P.; Romatschke, U. Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC? Phys. Rev. Lett.
**2007**, 99, 172301. [Google Scholar] [CrossRef] [Green Version] - Ryu, S.; Paquet, J.-F.; Shen, C.; Denicol, G.S.; Schenke, B.; Jeon, S.; Gale, C. Importance of the Bulk Viscosity of QCD in Ultrarelativistic Heavy-Ion Collisions. Phys. Rev. Lett.
**2015**, 115, 132301. [Google Scholar] [CrossRef] [PubMed] - Heinz, U.; Snellings, R. Collective flow and viscosity in relativistic heavy-ion collisions. Annu. Rev. Nucl. Part. Sci.
**2013**, 63, 123. [Google Scholar] [CrossRef] [Green Version] - Matsui, T.; Satz, H. J/psi Suppression by Quark-Gluon Plasma Formation. Phys. Lett. B
**1986**, 178, 416. [Google Scholar] [CrossRef] - Brambilla, N.; Pineda, A.; Soto, J.; Vairo, A. Effective field theories for heavy quarkonium. Rev. Mod. Phys.
**2005**, 77, 1423. [Google Scholar] [CrossRef] [Green Version] - Bazavov, A.; Petreczky, P.; Velytsky, A. Quarkonium at Finite Temperature. Quark-Gluon Plasma
**2010**, 4, 61. [Google Scholar] - Kaczmarek, O.; Karsch, F.; Petreczky, P.; Zantow, F. Heavy Quark Anti-Quark Free Energy and the Renormalized Polyakov Loop. Phys. Lett. B
**2002**, 543, 41. [Google Scholar] [CrossRef] [Green Version] - Torrieri, G.; Noronha, J. Flavoring the Quark-Gluon Plasma with Charm. Phys. Lett. B
**2010**, 690, 477–482. [Google Scholar] [CrossRef] [Green Version] - Bialas, A. Fluctuations of string tension and transverse mass distribution. Phys. Lett. B
**1999**, 466, 301–304. [Google Scholar] [CrossRef] [Green Version] - Steinke, S.; Rafelski, J. Quantum collective QCD string dynamics. J. Phys. G
**2006**, 32, S455–S460. [Google Scholar] [CrossRef] - Shifman, M.A.; Vainshtein, A.I.; Zakharov, V.I. QCD and Resonance Physics: Sum Rules. Nucl. Phys. B
**1979**, 147, 385. [Google Scholar] [CrossRef] - Shifman, M.A.; Vainshtein, A.I.; Zakharov, V.I. QCD and Resonance Physics: Applications. Nucl. Phys. B
**1979**, 147, 448. [Google Scholar] [CrossRef] - Romatschke, P.; Son, D.T. Spectral sum rules for the quark-gluon plasma. Phys. Rev. D
**2009**, 80, 065021. [Google Scholar] [CrossRef] - Meyer, H.B. Finite Temperature Sum Rules in Lattice Gauge Theory. Nucl. Phys. B
**2008**, 795, 230. [Google Scholar] [CrossRef] [Green Version] - Michael, C. Lattice Sum Rules. Nucl. Phys. B
**1987**, 280, 13. [Google Scholar] [CrossRef] [Green Version] - Rothe, H.J. A Novel Look at the Michael Lattice Sum Rules. Phys. Lett. B
**1995**, 355, 260. [Google Scholar] [CrossRef] [Green Version] - Michael, C. Lattice sum rules for the color fields. Phys. Rev. D
**1996**, 53, 4102. [Google Scholar] [CrossRef] [Green Version] - Rothe, H.J. Lattice energy sum rule and the trace anomaly. Phys. Lett. B
**1995**, 364, 227. [Google Scholar] [CrossRef] [Green Version] - Feuerbacher, B. Perturbative check of the energy lattice sum rule. Nucl. Phys. B
**2003**, 674, 484. [Google Scholar] [CrossRef] - Agasian, N.O. Low temperature relations in QCD. Phys. Atom. Nucl.
**2004**, 67, 391–395. [Google Scholar] [CrossRef] [Green Version] - Agasian, N.O. Nonperturbative vacuum and condensates in QCD below thermal phase transition. Phys. Lett. B
**2001**, 519, 71–77. [Google Scholar] [CrossRef] - Dosch, H.G.; Nachtmann, O.; Rueter, M. String Formation in the Model of the Stochastic Vacuum and Consistency with Low-Energy Theorems. arXiv
**1995**, arXiv:hep-ph/9503386. [Google Scholar] - Boyd, G.; Engels, J.; Karsch, F.; Laermann, E.; Legeland, C.; Lutgemeier, M.; Petersson, B. Thermodynamics of SU(3) Lattice Gauge Theory. Nucl. Phys. B
**1996**, 469, 419. [Google Scholar] [CrossRef] [Green Version] - Engels, J.; Fingberg, J.; Redlich, K.; Satz, H.; Weber, M. The Onset of Deconfinement in SU(2) Lattice Gauge Theory. Z. Phys. C
**1989**, 42, 341. [Google Scholar] [CrossRef] - Engels, J.; Karsch, F.; Satz, H.; Montvay, I. High Temperature SU(2) Gluon Matter on the Lattice. Phys. Lett. B
**1981**, 101, 89. [Google Scholar] [CrossRef] [Green Version] - Karsch, F. SU(N) Gauge Theory Couplings on Asymmetric Lattices. Nucl. Phys. B
**1982**, 205, 285. [Google Scholar] [CrossRef] [Green Version] - Bachas, C. Concavity of the Quarkonium Potential. Phys. Rev. D
**1986**, 33, 2723. [Google Scholar] [CrossRef] - Casimir, H.B.G. On the Attraction Between Two Perfectly Conducting Plates. Proc. Kom. Ned. Akad. Wetensch. B
**1948**, 51, 793. [Google Scholar] - Casimir, H.B.G.; Polder, D. The Influence of Retardation on the London-van der Waals Forces. Phys. Rev.
**1948**, 73, 360. [Google Scholar] [CrossRef] - Lee, I.; Park, K.; Lee, J. Precision density and volume contraction measurements of ethanol–water binary mixtures using suspended microchannel resonators. Sens. Actuators A
**2013**, 194, 62. [Google Scholar] [CrossRef] - Noronha, J. The Heavy Quark Free Energy in QCD and in Gauge Theories with Gravity Duals. Phys. Rev. D
**2010**, 82, 065016. [Google Scholar] [CrossRef] [Green Version] - Kaczmarek, O.; Zantow, F. Static quark anti-quark interactions in zero and finite temperature QCD. I: Heavy quark free energies, running coupling and quarkonium binding. Phys. Rev. D
**2005**, 71, 114510. [Google Scholar] [CrossRef] [Green Version] - Kaczmarek, O.; Karsch, F.; Laermann, E.; Lutgemeier, M. Heavy quark potentials in quenched QCD at high temperature. Phys. Rev. D
**2000**, 62, 034021. [Google Scholar] [CrossRef] [Green Version] - Laine, M.; Philipsen, O.; Romatschke, P.; Tassler, M. Real-time static potential in hot QCD. JHEP
**2007**, 03, 054. [Google Scholar] [CrossRef] [Green Version] - Wolschin, G. Aspects of Relativistic Heavy-Ion Collisions. Universe
**2020**, 6, 61. [Google Scholar] [CrossRef] - Bazavov, A.; Weber, J.H. Color Screening in Quantum Chromodynamics. Prog. Part. Nucl. Phys.
**2020**, 2020, 103823. [Google Scholar] [CrossRef] - Mocsy, A.; Petreczky, P. Heavy quarkonia survival in potential model. Eur. Phys. J. C
**2005**, 43, 77. [Google Scholar] [CrossRef] [Green Version] - Mateu, V.; Ortega, P.G.; Entem, D.R.; Fernández, F. Calibrating the Naïve Cornell Model with NRQCD. Eur. Phys. J. C
**2019**, 79, 323. [Google Scholar] [CrossRef]

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Chernodub, M.N.
Conformal Anomaly in Yang-Mills Theory and Thermodynamics of Open Confining Strings. *Universe* **2020**, *6*, 202.
https://doi.org/10.3390/universe6110202

**AMA Style**

Chernodub MN.
Conformal Anomaly in Yang-Mills Theory and Thermodynamics of Open Confining Strings. *Universe*. 2020; 6(11):202.
https://doi.org/10.3390/universe6110202

**Chicago/Turabian Style**

Chernodub, Maxim N.
2020. "Conformal Anomaly in Yang-Mills Theory and Thermodynamics of Open Confining Strings" *Universe* 6, no. 11: 202.
https://doi.org/10.3390/universe6110202