Dynamical symmetry breaking in hyperbolic 4D spacetime and in extra dimensions

D. Ebert ; A. V. Tyukov ; V. Ch. Zhukovsky

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Problems of modern theoretical and mathematical physics: Gauge theories and superstrings, Tome 272 (2011), pp. 97-116.

Résumé

We study the dynamical symmetry breaking in quark matter within two different models. First, we consider the effect of gravitational catalysis of chiral and color symmetries breaking in strong gravitational field of ultrastatic hyperbolic spacetime $\mathbb R\otimes H^3$ in the framework of an extended Nambu–Jona-Lasinio model. Second, we discuss the dynamical fermion mass generation in the flat 4-dimensional brane situated in the 5D spacetime with one extra dimension compactified on a circle. In the model, bulk fermions interact with fermions on the brane in the presence of a constant abelian gauge field $A_5$ in the bulk. The influence of the $A_5$-gauge field on the symmetry breaking is considered both when this field is a background parameter and a dynamical variable.

Bibliographie

[1] Nambu Y., Jona-Lasinio G., “Dynamical model of elementary particles based on an analogy with superconductivity. I”, Phys. Rev., 122 (1961), 345 | DOI

[2] Nambu Y., Jona-Lasinio G., “Dynamical model of elementary particles based on an analogy with superconductivity. II”, Phys. Rev., 124 (1961), 246 | DOI

[3] Vaks V.G., Larkin A.I., “O primenenii metodov teorii sverkhprovodimosti k voprosu o massakh elementarnykh chastits”, ZhETF, 40:1 (1961), 282 | Zbl

[4] Volkov M.K., Ebert D., “Chetyrekhkvarkovoe vzaimodeistvie kak obschii dinamicheskii istochnik modeli vektornoi dominantnosti i $\sigma $-modeli”, Yad. fiz., 36 (1982), 1265

[5] Ebert D., Volkov M.K., “Composite-meson model with vector dominance based on $U(2)$ invariant four-quark interactions”, Ztschr. Phys. C, 16 (1983), 205 | DOI

[6] Ebert D., Reinhardt H., “Effective chiral hadron Lagrangian with anomalies and Skyrme terms from quark flavour dynamics”, Nucl. Phys. B, 271 (1986), 188 | DOI

[7] Ebert D., Reinhardt H., Volkov M.K., “Effective hadron theory of QCD”, Prog. Part. and Nucl. Phys., 33 (1994), 1 | DOI

[8] Hatsuda T., Kunihiro T., “QCD phenomenology based on a chiral effective Lagrangian”, Phys. Rep., 247 (1994), 221 | DOI

[9] Ebert D., Kaschluhn L., Kastelewicz G., “Effective meson–diquark Lagrangian and mass formulas from the Nambu–Jona-Lasinio model”, Phys. Lett. B, 264 (1991), 420 | DOI

[10] Vogl U., “Diquarks from a $U(3)_L\times U(3)_R$ invariant quark Lagrangian”, Ztschr. Phys. A, 337 (1990), 191

[11] Vogl U., Weise W., “The Nambu and Jona-Lasinio model: Its implications for hadrons and nuclei”, Prog. Part. and Nucl. Phys., 27 (1991), 195 | DOI

[12] Barrois B.C., “Superconducting quark matter”, Nucl. Phys. B, 129 (1977), 390 | DOI

[13] Frautschi S.C., “Asymptotic freedom and color superconductivity in dense quark matter”, Hadronic matter at extreme energy density, Proc. Workshop, Erice (Italy), 1978, Plenum Press, New York, 1980, 18–27

[14] Bailin D., Love A., “Superfluidity and superconductivity in relativistic fermion systems”, Phys. Rep., 107 (1984), 325 | DOI

[15] Alford M., Rajagopal K., Wilczek F., “Color-flavor locking and chiral symmetry breaking in high density QCD”, Nucl. Phys. B, 537 (1999), 443 | DOI

[16] Langfeld K., Rho M., “Quark condensation, induced symmetry breaking and color superconductivity at high density”, Nucl. Phys. A, 660 (1999), 475 | DOI

[17] Berges J., Rajagopal K., “Color superconductivity and chiral symmetry restoration at non-zero baryon density and temperature”, Nucl. Phys. B, 538 (1999), 215 | DOI

[18] Schwarz T.M., Klevansky S.P., Papp G., “Phase diagram and bulk thermodynamical quantities in the Nambu–Jona-Lasinio model at finite temperature and density”, Phys. Rev. C, 60 (1999), 055205 | DOI

[19] Alford M., “Color-superconducting quark matter”, Ann. Rev. Nucl. and Part. Sci., 51 (2001), 131 | DOI

[20] Kerbikov B.O., Color superconducting state of quarks, E-print, 2001, arXiv: hep-ph/0110197

[21] Alford M.G., Schmitt A., Rajagopal K., Schäfer T., “Color superconductivity in dense quark matter”, Rev. Mod. Phys., 80 (2008), 1455 | DOI

[22] Shovkovy I.A., “Two lectures on color superconductivity”, Found. Phys., 35 (2005), 1309 | DOI | MR | Zbl

[23] Klimenko K.G., “Trekhmernaya model Grossa–Neve pri nenulevoi temperature i vo vneshnem magnitnom pole”, TMF, 90:1 (1992), 3 | MR

[24] Gusynin V.P., Miransky V.A., Shovkovy I.A., “Catalysis of dynamical flavor symmetry breaking by a magnetic field in $2+1$ dimensions”, Phys. Rev. Lett., 73 (1994), 3499 | DOI

[25] Gusynin V.P., Miransky V.A., Shovkovy I., “Dynamical flavor symmetry breaking by a magnetic field in $2+1$ dimensions”, Phys. Rev. D, 52 (1995), 4718 | DOI

[26] Gusynin V.P., Miransky V.A., Shovkovy I.A., “Dimensional reduction and dynamical chiral symmetry breaking by a magnetic field in $3+1$ dimensions”, Phys. Lett. B, 349 (1995), 477 | DOI

[27] Gusynin V.P., Miransky V.A., Shovkovy I.A., “Dynamical chiral symmetry breaking by a magnetic field in QED”, Phys. Rev. D, 52 (1995), 4747 | DOI

[28] Gusynin V.P., Miransky V.A., Shovkovy I.A., “Dimensional reduction and catalysis of dynamical symmetry breaking by a magnetic field”, Nucl. Phys. B, 462 (1996), 249 | DOI

[29] Klimenko K.G., Magnitsky B.V., Vshivtsev A.S., “Three-dimensional $(\psi \bar \psi )^2$ model with an external non-Abelian field, temperature and a chemical potential”, Nuovo Cimento A, 107 (1994), 439 | DOI

[30] Ebert D., Zhukovsky V.Ch., “Chiral phase transitions in strong background fields at finite temperature and dimensional reduction”, Mod. Phys. Lett. A, 12 (1997), 2567 | DOI | Zbl

[31] Ebert D., Klimenko K.G., Toki H., Zhukovsky V.Ch., “Chromomagnetic catalysis of color superconductivity and dimensional reduction”, Prog. Theor. Phys., 106 (2001), 835 | DOI | Zbl

[32] Ebert D., Khudyakov V.V., Zhukovsky V.Ch., Klimenko K.G., “Influence of an external chromomagnetic field on color superconductivity”, Phys. Rev. D, 65 (2002), 054024 | DOI | MR

[33] Inagaki T., Muta T., Odintsov S.D., “Dynamical symmetry breaking in curved spacetime: Four-fermion interactions”, Prog. Theor. Phys. Suppl., 127 (1997), 93 | DOI

[34] Gorbar E.V., “Dynamical symmetry breaking in spaces with a constant negative curvature”, Phys. Rev. D, 61 (1999), 024013 | DOI

[35] Gorbar E.V., “On effective dimensional reduction in hyperbolic spaces”, Ukr. J. Phys., 54:6 (2009), 541

[36] Gorbar E.V., Gusynin V.P., “Gap generation for Dirac fermions on Lobachevsky plane in a magnetic field”, Ann. Phys., 323 (2008), 2132 | DOI | Zbl

[37] Ebert D., Tyukov A.V., Zhukovsky V.Ch., “Gravitational catalysis of chiral and color symmetry breaking of quark matter in hyperbolic space”, Phys. Rev. D, 80 (2009), 085019 | DOI

[38] Antoniadis I., “A possible new dimension at a few TeV”, Phys. Lett. B, 246 (1990), 377 | DOI | MR

[39] Antoniadis I., Benakli K., Quiros M., “Production of Kaluza–Klein states at future colliders”, Phys. Lett. B, 331 (1994), 313 | DOI | MR

[40] Arkani-Hamed N., Dimopoulos S., Dvali G., “The hierarchy problem and new dimensions at a millimeter”, Phys. Lett. B, 429 (1998), 263 | DOI | MR

[41] Arkani-Hamed N., Dimopoulos S., Dvali G., “Phenomenology, astrophysics, and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity”, Phys. Rev. D, 59 (1999), 086004 | DOI

[42] Antoniadis I., Arkani-Hamed N., Dimopoulos S., Dvali G., “New dimensions at a millimeter to a Fermi and superstrings at a TeV”, Phys. Lett. B, 436 (1998), 257 | DOI

[43] Abe H., Miguchi H., Muta T., “Dynamical fermion masses under the influence of Kaluza–Klein fermions in extra dimensions”, Mod. Phys. Lett. A, 15 (2000), 445 | DOI

[44] Chang S., Hisano J., Nakano H., Okada N., Yamaguchi M., “Bulk standard model in the Randall–Sundrum background”, Phys. Rev. D, 62 (2000), 084025 | DOI | MR

[45] Han T., Lykken J.D., Zhang R.-J., “On Kaluza–Klein states from large extra dimensions”, Phys. Rev. D, 59 (1999), 105006 | DOI | MR

[46] Dobrescu B.A., “Electroweak symmetry breaking as a consequence of compact dimensions”, Phys. Lett. B, 461 (1999), 99 | DOI

[47] Cheng H.-C., Dobrescu B.A., Hill C.T., “Electroweak symmetry breaking and extra dimensions”, Nucl. Phys. B, 589 (2000), 249 | DOI

[48] Kobakhidze A.B., “Top-quark mass in the minimal top-condensation model with extra dimensions”, Yad. fiz., 64:5 (2001), 1010

[49] Manton N.S., “A new six-dimensional approach to the Weinberg–Salam model”, Nucl. Phys. B, 158 (1979), 141 | DOI | MR

[50] Fairlie D.B., “Higgs fields and the determination of the Weinberg angle”, Phys. Lett. B, 82 (1979), 97 | DOI

[51] Fairlie D.B., “Two consistent calculations of the Weinberg angle”, J. Phys. G, 5 (1979), L55–L58 | DOI

[52] Forgács P., Manton N.S., “Space–time symmetries in gauge theories”, Commun. Math. Phys., 72 (1980), 15 | DOI | MR

[53] Randjbar-Daemi S., Salam A., Strathdee J.A., “Spontaneous compactification in six-dimensional Einstein–Maxwell theory”, Nucl. Phys. B, 214 (1983), 491 | DOI | MR

[54] Sundrum R., To the fifth dimension and back, E-print, 2005, arXiv: hep-th/0508134

[55] Hosotani Y., “Dynamical mass generation by compact extra dimensions”, Phys. Lett. B, 126 (1983), 309 | DOI

[56] Hosotani Y., “Dynamics of non-integrable phases and gauge symmetry breaking”, Ann. Phys., 190 (1989), 233 | DOI | MR

[57] Parker L., Toms D.J., “Renormalization-group analysis of grand unified theories in curved spacetime”, Phys. Rev. D., 29 (1984), 1584 | DOI

[58] Brill D.R., Wheeler J.A., “Interaction of neutrinos and gravitational fields”, Rev. Mod. Phys., 29 (1957), 465 | DOI | MR | Zbl

[59] Dowker J.S., Apps J.S., Kirsten K., Bordag M., “Spectral invariants for the Dirac equation on the $d$-ball with various boundary conditions”, Class. and Quantum Grav., 13 (1996), 2911 | DOI | MR | Zbl

[60] Camporesi R., Higuchi A., “On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces”, J. Geom. and Phys., 20 (1996), 1 | DOI | MR | Zbl

[61] Ebert D., Tyukov A.V., Zhukovsky V.Ch., “Dynamical breaking and restoration of chiral and color symmetries in the static Einstein universe”, Phys. Rev. D, 76 (2007), 064029 | DOI | MR

[62] Bytsenko A.A., Cognola G., Vanzo L., Zerbini S., “Quantum fields and extended objects in space–times with constant curvature spatial section”, Phys. Rep., 266 (1996), 1 | DOI | MR

[63] Candelas P., Weinberg S., “Calculation of gauge couplings and compact circumferences from self-consistent dimensional reduction”, Nucl. Phys. B, 237 (1984), 397 | DOI | MR