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Form invariance, topological fluctuations and mass gap of Yang–Mills theory

Yachao Qian

Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA

E-mail Address: yachao.qian@alumni.stonybrook.edu

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and

Jun Nian

http://orcid.org/0000-0001-9294-2924

Institut des Hautes Édutes Scientifiques, Le Bois-Marie, 35 route de Chartres, 91440 Bures-sur-Yvette, France

C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794-3840, USA

E-mail Address: nian@ihes.fr

Corresponding author.

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https://doi.org/10.1142/S0217751X19501884Cited by:2 (Source: Crossref)

Abstract

In order to have a new perspective on the long-standing problem of the mass gap in Yang–Mills theory, we study in this paper the quantum Yang–Mills theory in the presence of topologically nontrivial backgrounds. The topologically stable gauge fields are constrained by the form invariance condition and the topological properties. Obeying these constraints, the known classical solutions to the Yang–Mills equation in the three- and four-dimensional Euclidean spaces are recovered, and the other allowed configurations form the nontrivial topological fluctuations at quantum level. Together, they constitute the background configurations, upon which the quantum Yang–Mills theory can be constructed. We demonstrate that the theory mimics the Higgs mechanism in a certain limit and develops a mass gap at semiclassical level on a flat space with finite size or on a sphere.

Keywords:

PACS: 11.10.Cd, 11.10.Lm, 11.15.Kc, 11.15.Yc, 11.27.+d

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References

1. C.-N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954). Web of Science, ADS, Google Scholar
2. S. L. Glashow, Nucl. Phys. 22, 579 (1961). Web of Science, Google Scholar
3. S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). Web of Science, ADS, Google Scholar
4. A. Salam, Conf. Proc. C680519, 367 (1968). Google Scholar
5. L. D. Faddeev and V. N. Popov, Phys. Lett. B 25, 29 (1967). Web of Science, ADS, Google Scholar
6. G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 44, 189 (1972). Web of Science, ADS, Google Scholar
7. A. A. Belavin, A. M. Polyakov, A. S. Schwartz and Yu. S. Tyupkin, Phys. Lett. B 59, 85 (1975). Web of Science, ADS, Google Scholar
8. T. T. Wu and C.-N. Yang, Some solutions of the classical isotopic gauge field equations, in Properties of Matter Under Unusual Conditions (Interscience Publishers, 1969), p. 349. Google Scholar
9. E. Witten, Phys. Rev. Lett. 38, 121 (1977). Web of Science, ADS, Google Scholar
10. A. Actor, Rev. Mod. Phys. 51, 461 (1979). Web of Science, ADS, Google Scholar
11. N. Seiberg and E. Witten, Nucl. Phys. B 426, 19 (1994) [Erratum: ibid. 430, 485 (1994)], arXiv:hep-th/9407087. Web of Science, ADS, Google Scholar
12. A. Jaffe and E. Witten, Quantum Yang–Mills theory, The Millennium Prize Problems (Clay Mathematics Institute and American Mathematical Society, 2006), pp. 129–152. Google Scholar
13. P. W. Anderson, Phys. Rev. 130, 439 (1963). Web of Science, ADS, Google Scholar
14. F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964). Web of Science, ADS, Google Scholar
15. P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964). Web of Science, ADS, Google Scholar
16. G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1964). Web of Science, ADS, Google Scholar
17. R. P. Feynman, Phys. Rev. 97, 660 (1955). Web of Science, ADS, Google Scholar
18. A. M. Polyakov, Nucl. Phys. B 120, 429 (1977). Web of Science, ADS, Google Scholar
19. R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That (Addison-Wesley, 1989). Google Scholar
20. J. P. Bourguignon, H. B. Lawson and J. Simons, Proc. Natl. Acad. Sci. 76, 1550 (1979). Web of Science, ADS, Google Scholar
21. A. Cucchieri and T. Mendes, PoS LAT2007, 297 (2007), arXiv:0710.0412 [hep-lat]. Google Scholar
22. I. L. Bogolubsky, E. M. Ilgenfritz, M. Muller-Preussker and A. Sternbeck, PoS LAT2007, 290 (2007), arXiv:0710.1968 [hep-lat]. Google Scholar
23. O. Oliveira, P. J. Silva, E. M. Ilgenfritz and A. Sternbeck, PoS LAT2007, 323 (2007), arXiv:0710.1424 [hep-lat]. Google Scholar
24. M. Q. Huber, A. Maas and L. von Smekal, J. High Energy Phys. 11, 035 (2012), arXiv:1207.0222 [hep-th]. Web of Science, ADS, Google Scholar
25. S. Dubovsky, R. Flauger and V. Gorbenko, Phys. Rev. Lett. 111, 062006 (2013), arXiv:1301.2325 [hep-th]. Web of Science, ADS, Google Scholar
26. S. Dubovsky, R. Flauger and V. Gorbenko, J. Exp. Theor. Phys. 120, 399 (2015), arXiv:1404.0037 [hep-th]. Web of Science, ADS, Google Scholar
27. D. Diakonov and V. Yu. Petrov, Nucl. Phys. B 245, 259 (1984). Web of Science, ADS, Google Scholar
28. S. Mazurkiewicz, Studia Mathematica 3, 92 (1931). Google Scholar
29. S. Banach, Studia Mathematica 3, 174 (1931). Google Scholar
30. R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948). Web of Science, ADS, Google Scholar
31. P. Gérard, Annales de l’I.H.P. Analyse non linéaire 23, 765 (2006). Web of Science, ADS, Google Scholar
32. P. Gérard, The Gross-Pitaevskii equation in the energy space, in Stationary and Time Dependent Gross–Pitaevskii Equations, Contemporary Mathematics, Vol. 473 (American Mathematical Society, 2008). Google Scholar
33. R. Killip, T. Oh, O. Pocovnicu and M. Visan, Math. Res. Lett. 19, 969 (2012), arXiv:1112.1354 [math.AP]. Google Scholar
34. W. Bao and Y. Cai, Kinetic Related Models 6, 1 (2013), arXiv:1212.5341 [cond-mat.quant-gas]. Web of Science, Google Scholar
35. E. H. Lieb, R. Seiringer and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross–Pitaevskii energy functional, in The Stability of Matter : From Atoms to Stars : Selecta of Elliott H. Lieb (Springer, Berlin, 2005), pp. 759–771. Google Scholar
36. M. I. Weinstein, Commun. Math. Phys. 87, 567 (1982). Web of Science, ADS, Google Scholar
37. W. Bao and X. Ruan, Fundamental gaps and energy asymptotics of the Gross–Pitaevskii/nonlinear Schrödinger equation with repulsive interaction, arXiv:1512.07123 [math-ph]. Google Scholar
38. J. Nian and Y. Qian, A topological way of finding solutions to Yang–Mills equation, arXiv:1901.06818 [hep-th]. Google Scholar
39. D. Weingarten, Nucl. Phys. B (Proc. Suppl.) 34, 29 (1994), arXiv:hep-lat/9401021. ADS, Google Scholar
40. C. J. Morningstar and M. J. Peardon, Phys. Rev. D 60, 034509 (1999), arXiv:hep-lat/9901004. Web of Science, ADS, Google Scholar
41. B. Lucini, M. Teper and U. Wenger, J. High Energy Phys. 06, 012 (2004), arXiv:hep-lat/0404008. Web of Science, ADS, Google Scholar
42. Y. Chen et al., Phys. Rev. D 73, 014516 (2006), arXiv:hep-lat/0510074. Web of Science, ADS, Google Scholar
43. W. Ochs, J. Phys. G 40, 043001 (2013), arXiv:1301.5183 [hep-ph]. Web of Science, Google Scholar
44. C.-H. Gu, Phys. Rep. 80, 251 (1981). Web of Science, Google Scholar
45. I. Zahed and G. E. Brown, Phys. Rep. 142, 1 (1986). Web of Science, ADS, Google Scholar
46. J. J. Oh, C. Park and H. S. Yang, J. High Energy Phys. 04, 087 (2011), arXiv:1101.1357 [hep-th]. Web of Science, ADS, Google Scholar
47. Z.-Q. Ma and B.-W. Xu, J. Phys. A 17, L389 (1984). Google Scholar
48. Z.-Q. Ma and B.-W. Xu, J. Phys. A 17, L719 (1984). Google Scholar
49. S. Vandoren and P. van Nieuwenhuizen, Lectures on instantons, arXiv:0802.1862 [hep-th]. Google Scholar
50. E. Fradkin, Field Theories of Condensed Matter Systems (Addison-Wesley, 1991). Google Scholar