Abstract
In this paper quantum effects are investigated in a very special two-scalar field model having a moduli space of BPS topological defects. In a (1 + 1)-dimensional space-time the defects are classically degenerate in mass kinks, but in (3 + 1) dimensions the kinks become BPS domain walls, all of them sharing the same surface tension at the classical level. The heat kernel/zeta function regularization method will be used to control the divergences induced by the quantum kink and domain wall fluctuations. A generalization of the Gilkey-DeWitt-Avramidi heat kernel expansion will be developed in order to accommodate the infrared divergences due to zero modes in the spectra of the second-order kink and domain wall fluctuation operators, which are respectively N = 2 × N = 2 matrix ordinary or partial differential operators. Use of these tools in the spectral zeta function associated with the Hessian operators paves the way to obtain general formulas for the one-loop kink mass and domain wall tension shifts in any (1 + 1)- or (3 + 1)-dimensional N -component scalar field theory model. Application of these formulae to the BPS kinks or domain walls of the N = 2 model mentioned above reveals the breaking of the classical mass or surface tension degeneracy at the quantum level. Because the main parameter distinguishing each member in the BPS kink or domain wall moduli space is essentially the distance between the centers of two basic kinks or walls, the breaking of the degeneracy amounts to the surge in quantum-induced forces between the two constituent topological defects. The differences in surface tension induced by one-loop fluctuations of BPS walls give rise mainly to attractive forces between the constituent walls except if the two basic walls are very far apart. Repulsive forces between two close walls only arise if the coupling approaches the critical value from below.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Vilenkin and E.P.S. Shellard, Cosmic strings and other topological defects, Cambridge University Press, Cambridge U.K. (1994).
M.A. Shifman and M. Voloshin, Degenerate domain wall solutions in supersymmetric theories, Phys. Rev. D 57 (1998) 2590 [hep-th/9709137] [INSPIRE].
M. Eto and N. Sakai, Solvable models of domain walls in N = 1 supergravity, Phys. Rev. D 68 (2003) 125001 [hep-th/0307276] [INSPIRE].
D. Bazeia, J. Nascimento, R. Ribeiro and D. Toledo, Soliton stability in systems of two real scalar fields, J. Phys. A 30 (1997) 8157 [hep-th/9705224] [INSPIRE].
A.A. Izquierdo, M. González Leon and J.M. Guilarte, The kink variety in systems of two coupled scalar fields in two space-time dimensions, Phys. Rev. D 65 (2002) 085012 [hep-th/0201200] [INSPIRE].
N. Manton, A remark on the scattering of BPS monopoles, Phys. Lett. B 110 (1982) 54 [INSPIRE].
N. Manton and P. Sutcliffe, Topological solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2004).
A. Alonso Izquierdo, M. González Leon, J. Mateos Guilarte and M. de la Torre Mayado, Adiabatic motion of two component BPS kinks, Phys. Rev. D 66 (2002) 105022 [hep-th/0207064] [INSPIRE].
D. Tong, The moduli space of BPS domain walls, Phys. Rev. D 66 (2002) 025013 [hep-th/0202012] [INSPIRE].
N.D. Antunes, E.J. Copeland, M. Hindmarsh and A. Lukas, Kinky brane worlds, Phys. Rev. D 68 (2003) 066005 [hep-th/0208219] [INSPIRE].
R.F. Dashen, B. Hasslacher and A. Neveu, Nonperturbative methods and extended hadron models in field theory. 2. Two-dimensional models and extended hadrons, Phys. Rev. D 10 (1974) 4130 [INSPIRE].
A.S. Goldhaber, A. Rebhan, P. van Nieuwenhuizen and R. Wimmer, Quantum corrections to mass and central charge of supersymmetric solitons, Phys. Rept. 398 (2004) 179 [hep-th/0401152] [INSPIRE].
M.A. Shifman, A.I. Vainshtein and M.B. Voloshin, Anomaly and quantum corrections to solitons in two-dimensional theories with minimal supersymmetry, Phys. Rev. D 59 (1999) 045016 [hep-th/9810068] [INSPIRE].
N. Graham and R. Jaffe, Energy, central charge and the BPS bound for (1 + 1)-dimensional supersymmetric solitons, Nucl. Phys. B 544 (1999) 432 [hep-th/9808140] [INSPIRE].
A. Rebhan and P. van Nieuwenhuizen, No saturation of the quantum Bogomolnyi bound by two-dimensional supersymmetric solitons, Nucl. Phys. B 508 (1997) 449 [hep-th/9707163] [INSPIRE].
H. Nastase, M.A. Stephanov, P. van Nieuwenhuizen and A. Rebhan, Topological boundary conditions, the BPS bound and elimination of ambiguities in the quantum mass of solitons, Nucl. Phys. B 542 (1999) 471 [hep-th/9802074] [INSPIRE].
A. Alonso Izquierdo, W. Garcia Fuertes, M. Gonzalez Leon and J. Mateos Guilarte, One loop correction to classical masses of quantum kink families, Nucl. Phys. B 681 (2004) 163 [hep-th/0304125] [INSPIRE].
J. Mateos Guilarte, A. Alonso-Izquierdo, W. Garcia Fuertes, M. de la Torre Mayado and M. Senosiain, Quantum fluctuations around low-dimensional topological defects, PoS(ISFTG)013 [arXiv:0909.2107] [INSPIRE].
M. Bordag and J. Muñoz-Castañeda, Quantum vacuum interaction between two sine-Gordon kinks, J. Phys. A 45 (2012) 374012 [arXiv:1112.3237] [INSPIRE].
J.M. Muñoz-Castañeda, J.M. Guilarte and A.M. Mosquera, Quantum vacuum energies and Casimir forces between partially transparent δ-function plates, Phys. Rev. D 87 (2013) 105020 [arXiv:1305.2054] [INSPIRE].
E. Elizalde, S. Odintsov, A. Romeo, A. Bytsenko and S. Zerbini, Zeta regularization techniques with applications, World Scientific, Singapore (1994).
K. Kirsten, Spectral functions in mathematics and physics, Chapman and Hall/CRC, New York U.S.A. (2002).
D. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
B.S. de Witt, Dynamical theory of groups and fields, Gordon and Breach, U.S.A. (1965).
P.B. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem, Publish or Perish Inc, U.S.A. (1984).
J. Roe, Elliptic operators, topology and asymptotic methods, Longman Scientific and Technical, New York U.S.A. (1988).
I.G. Avramidi, Heat kernel approach in quantum field theory, Nucl. Phys. Proc. Suppl. 104 (2002) 3 [math-ph/0107018] [INSPIRE].
I. Avramidi and R. Schimming, Heat kernel coefficients to the matrix Schrödinger operator, J. Math. Phys. 36 (1995) 5042 [hep-th/9501026] [INSPIRE].
J. Dowker and R. Critchley, Effective Lagrangian and energy momentum tensor in de Sitter space, Phys. Rev. D 13 (1976) 3224 [INSPIRE].
S. Hawking, Zeta function regularization of path integrals in curved space-time, Commun. Math. Phys. 55 (1977) 133 [INSPIRE].
M. Bordag, A.S. Goldhaber, P. van Nieuwenhuizen and D. Vassilevich, Heat kernels and zeta function regularization for the mass of the SUSY kink, Phys. Rev. D 66 (2002) 125014 [hep-th/0203066] [INSPIRE].
A. Alonso Izquierdo, W. Garcia Fuertes, M. Gonzalez Leon and J. Mateos Guilarte, Generalized zeta functions and one loop corrections to quantum kink masses, Nucl. Phys. B 635 (2002) 525 [hep-th/0201084] [INSPIRE].
A. Alonso-Izquierdo, J.M. Guilarte and M.S. Plyushchay, Kink mass quantum shifts from SUSY quantum mechanics, Annals Phys. 331 (2013) 269 [arXiv:1212.0818] [INSPIRE].
A. Alonso Izquierdo, W. Garcia Fuertes, M. Gonzalez Leon and J. Mateos Guilarte, Semiclassical mass of quantum k component topological kinks, Nucl. Phys. B 638 (2002) 378 [hep-th/0205137] [INSPIRE].
A. Alonso-Izquierdo and J.M. Guilarte, One-loop kink mass shifts: a computational approach, Nucl. Phys. B 852 (2011) 696 [arXiv:1107.2216] [INSPIRE].
A. Alonso-Izquierdo and J. Mateos-Guilarte, Kink fluctuation asymptotics and zero modes, Eur. Phys. J. C 72 (2012) 2170 [arXiv:1207.0942] [INSPIRE].
A. Alonso-Izquierdo and J. Mateos-Guilarte, Gilkey-de Witt heat kernel expansion and zero modes, Nuovo Cim. C 3 (2013) 3 [arXiv:1303.3138] [INSPIRE].
C. Aragão de Carvalho, G. Marques, A. da Silva and I. Ventura, Domain walls at finite temperature, Nucl. Phys. B 265 (1986) 45 [INSPIRE].
A. Litvintsev and P. van Nieuwenhuizen, Once more on the BPS bound for the SUSY kink, hep-th/0010051 [INSPIRE].
A. Rebhan, P. van Nieuwenhuizen and R. Wimmer, One loop surface tensions of (supersymmetric) kink domain walls from dimensional regularization, New J. Phys. 4 (2002) 31 [hep-th/0203137] [INSPIRE].
A. Rebhan, A. Schmitt and P. van Nieuwenhuizen, One-loop results for kink and domain wall profiles at zero and finite temperature, Phys. Rev. D 80 (2009) 045012 [arXiv:0903.5242] [INSPIRE].
A. Alonso-Izquierdo, M.A. González León, J.M. Guilarte and M. de la Torre Mayado, On domain walls in a Ginzburg-Landau non-linear S 2 -σ-model, JHEP 08 (2010) 111 [arXiv:1009.0617] [INSPIRE].
A. Alonso-Izquierdo, M.A. González León and J. Mateos Guilarte, Kinks in a non-linear massive σ-model, Phys. Rev. Lett. 101 (2008) 131602 [arXiv:0808.3052] [INSPIRE].
A.A. Izquierdo, M.A. González León and J. Mateos Guilarte, BPS and non-BPS kinks in a massive non-linear S 2 -σ-model, Phys. Rev. D 79 (2009) 125003 [arXiv:0903.0593] [INSPIRE].
A. Alonso-Izquierdo and J. Mateos Guilarte, On a family of (1 + 1)-dimensional scalar field theory models: kinks, stability, one-loop mass shifts, Annals Phys. 327 (2012) 2251 [arXiv:1205.3069] [INSPIRE].
E. Megias, E. Ruiz Arriola and L. Salcedo, The thermal heat kernel expansion and the one loop effective action of QCD at finite temperature, Phys. Rev. D 69 (2004) 116003 [hep-ph/0312133] [INSPIRE].
A. Alonso Izquierdo, W. García Fuertes, M. de la Torre Mayado and J. Mateos Guilarte, Quantum corrections to the mass of self-dual vortices, Phys. Rev. D 70 (2004) 061702 [hep-th/0406129] [INSPIRE].
A. Alonso Izquierdo, W. García Fuertes, M. de la Torre Mayado and J. Mateos Guilarte, Quantum oscillations of self-dual Abrikosov-Nielsen-Olesen vortices, Phys. Rev. D 71 (2005) 125010 [hep-th/0504143] [INSPIRE].
A.A. Izquierdo, W. García Fuertes, M. de la Torre Mayado and J.M. Guilarte, One loop corrections to the mass of self-dual semi-local planar topological solitons, Nucl. Phys. B 797 (2008) 431 [arXiv:0707.4592] [INSPIRE].
G. Gibbons, M. Ortiz, F. Ruiz Ruiz and T. Samols, Semilocal strings and monopoles, Nucl. Phys. B 385 (1992) 127 [hep-th/9203023] [INSPIRE].
S.R. Coleman, The fate of the false vacuum. 1. Semiclassical theory, Phys. Rev. D 15 (1977) 2929 [Erratum ibid. D 16 (1977) 1248] [INSPIRE].
A. Wipf, Tunnel determinants, Nucl. Phys. B 269 (1986) 24 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1307.0740
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Alonso-Izquierdo, A., Guilarte, J.M. Quantum-induced interactions in the moduli space of degenerate BPS domain walls. J. High Energ. Phys. 2014, 125 (2014). https://doi.org/10.1007/JHEP01(2014)125
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2014)125