# Tachyons and Solitons in Spontaneous Symmetry Breaking in the Frame of Field Theory

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## Abstract

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## 1. Introduction

## 2. Hysteresis Phenomenon of the Unstable Critical Point in the Broken Symmetry Phase

## 3. Solitons and Tachyons in SSB

## 4. Where Are the Tachyons in the SSB Phase?

## 5. Discussion about Phenomena Within the Zone $\Delta T$

## 6. Discussion for Tachyons and Solitons at the Critical Point of QCD and Beyond

## 7. Conclusions—Further Investigations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Huang, K. Statistical Mechanics, 2nd ed.; Wiley: New York, NY, USA, 1987. [Google Scholar]
- Ryder, L.H. Quantum Field Theory; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
- Kaku, M. Quantum Field Theory: A Modern Introduction; Oxford University Press: New York, NY, USA, 1993. [Google Scholar]
- Felder, G.; Kofman, L.; Linde, A. Tachyonic instability and dynamics of spontaneous symmetry breaking. Phys. Rev. D
**2001**, 64, 123517. [Google Scholar] [CrossRef] [Green Version] - Felder, G.; Garcia-Bellido, J.; Greene, P.B.; Kofman, L.; Linde, A.; Tkachev, I. Dynamics of symmetry breaking and tachyonic preheating. Phys. Rev. Lett.
**2001**, 87, 011601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Armoni, A.; Lopez, E. UV/IR mixing via closed strings and tachyonic instabilities. Nucl. Phys. B
**2002**, 632, 240–256. [Google Scholar] [CrossRef] [Green Version] - Andreev, A.Y.; Kirzhnits, D.A. Tachyons and the instability of physical systems. Usp. Fiz. Nauk
**1996**, 166, 1135. [Google Scholar] [CrossRef] - De Alwis, S.P.; Flournoy, A.T. Closed string tachyons and semiclassical instabilities. Phys. Rev. D
**2002**, 66, 026005. [Google Scholar] [CrossRef] [Green Version] - Dymnikova, I. Mass, spacetime symmetry, de Sitter vacuum, and the Higgs mechanism. Symmetry
**2020**, 12, 634. [Google Scholar] [CrossRef] [Green Version] - Pollock, M.D. World-sheet stability, space-time horizons and cosmic censorship. Eur. Phys. J. Plus
**2014**, 129, 257. [Google Scholar] [CrossRef] - Sen, A. Uniqueness of Tachyonic Solitons. J. High Energy Phys.
**2001**, JHEP12, 001. [Google Scholar] [CrossRef] [Green Version] - Newman, M.E.J.; Barkema, G.T. Monte Carlo Methods in Statistical Mechanics; Oxford University Press: New York, NY, USA, 1999. [Google Scholar]
- Contoyiannis, Y.; Diakonos, F.; Malakis, A. Intermittent dynamics of critical fluctuations. Phys. Rev. Lett.
**2002**, 89, 35701. [Google Scholar] [CrossRef] [PubMed] - Contoyiannis, Y.; Potirakis, S.M. Signatures of the symmetry breaking phenomenon in pre-seismic electromagnetic emissions. J. Stat. Mech.
**2018**, 083208. [Google Scholar] [CrossRef] - Contoyiannis, Y.; Potirakis, S.M.; Stavrinides, S.G.; Hanias, M.P.; Tassis, D.; Theodorou, C.G. Intermittency-induced criticality in the random telegraph noise of nanoscale UTBB FD-SOI MOSFETs. Microelectron. Eng.
**2019**, 216, 111027. [Google Scholar] [CrossRef] - García-Etxebarria, I.; Montero, M.; Uranga, A.M. Closed tachyon solitons in type II string theory. Fortschr. Phys.
**2015**, 63, 571–595. [Google Scholar] [CrossRef] [Green Version] - Giaccari, S.; Nian, J. Dark solitons, D-branes and noncommutative tachyon field theory. Int. J. Modern Phys. A
**2017**, 32, 1750201. [Google Scholar] [CrossRef] [Green Version] - Dvali, G.; Gomez, C.; Gruending, L.; Rug, T. Towards a quantum theory of solitons. Nucl. Phys. B
**2015**, 901, 338–353. [Google Scholar] [CrossRef] [Green Version] - Diakonos, F.K.; Contoyiannis, Y.F.; Potirakis, S.M. Spontaneous symmetry breaking in finite systems and anomalous order-parameter correlations. arXiv
**2021**, arXiv:2104.11662v1. [Google Scholar] - Voloshin, M.B. The rate of metastable vacuum decay in (2 + 1) dimensions. Phys. Lett. B
**2004**, 599, 129–135. [Google Scholar] [CrossRef] [Green Version] - Wong, C.Y. Introduction to High-Energy Heavy-Ion Collisions; World Scientific Publishing: Singapore, 1994. [Google Scholar]
- Kogut, J.B. An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys.
**1979**, 51, 659. [Google Scholar] [CrossRef] - Balian, R.; Drouffe, J.M.; Itzykson, C. Gauge fields on a lattice. II. Gauge-invariant Ising model. Phys. Rev. D
**1975**, 11, 2098. [Google Scholar] [CrossRef] - Antoniou, N.; Contoyiannis, Y.; Diakonos, F.; Karanikas, A.; Ktorides, C. Pion production from a critical QCD phase. Nucl. Phys. A
**2001**, 693, 799–824. [Google Scholar] [CrossRef] [Green Version] - Issifu, A.; Brito, F.A. The (de)confinement transition in tachyonic matter at finite temperature. Adv. High Energy Phys.
**2019**, 9450367. [Google Scholar] [CrossRef] [Green Version] - Brito, F.A.; Freire, M.L.F.; Serafim, W. Confinement and screening in tachyonic matter. Eur. Phys. J. C
**2014**, 74, 3202. [Google Scholar] [CrossRef] [Green Version] - Shuryak, E.; Schaefer, T. Instantons and chiral symmetry restoration in QCD-like theories. Nucl. Phys. B Proc. Suppl.
**1997**, 53, 472–474. [Google Scholar] [CrossRef] - Escamilla-Rivera, C.; García-Jiménez, G.; Loaiza-Brito, O.; Obregón, O. Closed string tachyon: Inflation and cosmological collapse. Class. Quantum Grav.
**2013**, 30, 035005. [Google Scholar] [CrossRef] [Green Version] - Kofman, L.; Linde, A. Problems with tachyon inflation. J. High Energy Phys.
**2002**, 7, 004. [Google Scholar] [CrossRef] - Pourhassan, B.; Naji, J. Tachyonic matter cosmology with exponential and hyperbolic potentials. Int. J. Mod. Phys. D
**2017**, 26, 1750012. [Google Scholar] [CrossRef] - El-Nablusi, A.R. Phase transitions in the early universe with negatively induced supergravity cosmological constant. Chin. Phys. Lett.
**2006**, 23, 1124. [Google Scholar] [CrossRef] - El-Nabulsi, R.A. Effective cosmological constant from supergravity arguments and non-minimal coupling. Phys. Lett. B
**2005**, 619, 26–29. [Google Scholar] [CrossRef] - Contoyiannis, Y.; Stavrinides, S.G.; Kampitakis, M.; Hanias, M.P.; Potirakis, S.M.; Papadopoulos, P. Spontaneous symmetry breaking in the phase space. Phys. Scr.
**2021**, 96, 075204. [Google Scholar] [CrossRef]

**Figure 1.**Free energy $U\left(\phi \right)$ for the symmetric phase (green curve) and the SSB (blue curve) (also see text). The minima in blue curve are marked at ${\phi}^{*}=\pm \sqrt{\frac{{r}_{0}}{{u}_{0}}}$, for ${r}_{0}={u}_{0}=1$.

**Figure 2.**Invariant density diagrams for the 3D Ising numerical experiment (${20}^{3}$ lattice, ${N}_{iter}=200,000$ sweeps) for 4 different temperatures: (

**a**) the critical point at ${T}_{c}=4.545$. (

**b**) The two lobes at $T=4.48$ indicate the two stable fixed points. The separation of the lobes is not yet complete. This indicates the presence of the unstable critical point. (

**c**) At ${T}_{SSB}=4.44$, the SSB has been completed. This is the last presence of the unstable critical point. (

**d**) Below the SSB there is only one lobe; here, an example for $\mathrm{T}=4.42<{T}_{SSB}$ is shown. The unstable critical point does not exist anymore. The existence of this evolution has first been reported in [14], referring to the preparation of strong earthquake events, as well as in the random telegraph noise of nanoscale UTBB FD-SOI MOSFETs [15].

**Figure 3.**The finite size effect of the unstable critical point is present in the broken symmetry phase. The hysteresis zone $\Delta T$ values for the lattice lengths $L=8,10,13,16,20,23,26,30,34$ are presented. A power–law relationship between $\Delta T$ and $L$ of the form $\Delta T~{L}^{-1.5}$ exists. This means that in small distances the hysteresis phenomenon could be important.

**Figure 4.**The dot blue curve is produced if the SSB free energy ${U}_{SSB}\left(\phi \right)$ (blue curve in Figure 1) shifted by $U=\frac{{r}_{0}{}^{2}}{4{u}_{0}}$ (or $U=\frac{{m}^{4}}{4\lambda}$ for field theories); the green curve represents free energy $U\left(\phi \right)$ for the symmetric phase (as in Figure 1).

**Figure 6.**(

**a**) The kink soliton at ${\mathrm{T}}_{\mathrm{SSB}}$ in the magnetization of 3D Ising model localized in time. (

**b**) Within the hysteresis zone, $\left({\mathrm{T}}_{\mathrm{SSB}}<\mathrm{T}<{\mathrm{T}}_{\mathrm{c}}\right)$ the kink soliton recognizes two components which are expressed by the fluctuations within the new vacua, as in case (

**a**), and fluctuations around the critical point $M=0$. The red dashed horizontal line at $M=0$ is provided to guide the eye. The scales in y-axis as well as in x-axis are the same in both plots for comparison reasons (see text for interpretation of these fluctuations).

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**MDPI and ACS Style**

Contoyiannis, Y.; Hanias, M.P.; Papadopoulos, P.; Stavrinides, S.G.; Kampitakis, M.; Potirakis, S.M.; Balasis, G.
Tachyons and Solitons in Spontaneous Symmetry Breaking in the Frame of Field Theory. *Symmetry* **2021**, *13*, 1358.
https://doi.org/10.3390/sym13081358

**AMA Style**

Contoyiannis Y, Hanias MP, Papadopoulos P, Stavrinides SG, Kampitakis M, Potirakis SM, Balasis G.
Tachyons and Solitons in Spontaneous Symmetry Breaking in the Frame of Field Theory. *Symmetry*. 2021; 13(8):1358.
https://doi.org/10.3390/sym13081358

**Chicago/Turabian Style**

Contoyiannis, Yiannis, Michael P. Hanias, Pericles Papadopoulos, Stavros G. Stavrinides, Myron Kampitakis, Stelios M. Potirakis, and Georgios Balasis.
2021. "Tachyons and Solitons in Spontaneous Symmetry Breaking in the Frame of Field Theory" *Symmetry* 13, no. 8: 1358.
https://doi.org/10.3390/sym13081358