Abstract
In this work a lattice formulation of a supersymmetric theory is proposed and tested that preserves the complete supersymmetry on the lattice. The results of a onedimensional nonperturbative simulation show the realization of the full supersymmetry and the correct continuum limit of the theory. It is proven here that the violation of supersymmetry due to the absence of the Leibniz rule on the lattice can be amended only with a nonlocal derivative and nonlocal interaction term. The fermion doubling problem is also discussed, which leads to another important source of supersymmetry breaking on the lattice. This problem is also solved with a nonlocal realization.
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References
J. Giedt, Deconstruction and other approaches to supersymmetric lattice field theories, Int. J. Mod. Phys. A 21 (2006) 3039 [hep-lat/0602007] [SPIRES].
A. Feo, Supersymmetry on the lattice, Nucl. Phys. Proc. Suppl. 119 (2003) 198 [hep-lat/0210015] [SPIRES].
D.B. Kaplan, Supersymmetry on the lattice, Eur. Phys. J. ST 152 (2007) 89 [SPIRES].
I. Montvay, Tuning to N = 2 supersymmetry in the SU(2) adjoint Higgs-Yukawa model, Nucl. Phys. B 445 (1995) 399 [hep-lat/9503009] [SPIRES].
S. Elitzur, E. Rabinovici and A. Schwimmer, Supersymmetric models on the lattice, Phys. Lett. B 119 (1982) 165 [SPIRES].
S. Catterall and S. Karamov, A two-dimensional lattice model with exact supersymmetry, Nucl. Phys. Proc. Suppl. 106 (2002) 935 [hep-lat/0110071] [SPIRES].
D.B. Kaplan, E. Katz and M. Ünsal, Supersymmetry on a spatial lattice, JHEP 05 (2003) 037 [hep-lat/0206019] [SPIRES].
S. Catterall, D.B. Kaplan and M. Ünsal, Exact lattice supersymmetry, Phys. Rept. 484 (2009) 71 [arXiv:0903.4881] [SPIRES].
P.H. Dondi and H. Nicolai, Lattice supersymmetry, Nuovo Cim. A 41 (1977) 1 [SPIRES].
S. Nojiri, Continuous ‘translation’ and supersymmetry on the lattice, Prog. Theor. Phys. 74 (1985) 819 [SPIRES].
S. Nojiri, The spontaneous breakdown of supersymmetry on the finite lattice, Prog. Theor. Phys. 74 (1985) 1124 [SPIRES].
J. Bartels and G. Kramer, A lattice version of the Wess-Zumino model, Z. Phys. C 20 (1983) 159 [SPIRES].
M. Kato, M. Sakamoto and H. So, Taming the Leibniz rule on the lattice, JHEP 05 (2008) 057 [arXiv:0803.3121] [SPIRES].
G. Bergner, T. Kaestner, S. Uhlmann and A. Wipf, Low-dimensional supersymmetric lattice models, Annals Phys. 323 (2008) 946 [arXiv:0705.2212] [SPIRES].
G. Bergner, Symmetries an the methods of quantum field theory: supersymmetry on a space-time lattice, Ph.D. thesis, Friedrich-Schiller-Universität Jena Jena, Jena, Germany (2009), online at http://www.tpi.uni-jena.de/qfphysics/thesis/georg_bergner_phd.pdf.
J. Wess and B. Zumino, A lagrangian model invariant under supergauge transformations, Phys. Lett. B 49 (1974) 52 [SPIRES].
T. Kastner, G. Bergner, S. Uhlmann, A. Wipf and C. Wozar, Two-dimensional Wess-Zumino models at intermediate couplings, Phys. Rev. D 78 (2008) 095001 [arXiv:0807.1905] [SPIRES].
G. Bergner, F. Bruckmann and J.M. Pawlowski, Generalising the Ginsparg-Wilson relation: lattice supersymmetry from blocking transformations, Phys. Rev. D 79 (2009) 115007 [arXiv:0807.1110] [SPIRES].
A. D’Adda, N. Kawamoto and J. Saito, Formulation of supersymmetry on a lattice as a representation of a deformed superalgebra, arXiv:0907.4137 [SPIRES].
F. Bruckmann and M. de Kok, Noncommutativity approach to supersymmetry on the lattice: SUSY quantum mechanics and an inconsistency, Phys. Rev. D 73 (2006) 074511 [hep-lat/0603003] [SPIRES].
H.B. Nielsen and M. Ninomiya, No-Go theorem for regularizing chiral fermions, Phys. Lett. B 105 (1981) 219 [SPIRES].
D. Friedan, A proof of the Nielsen-Ninomiya theorem, Commun. Math. Phys. 85 (1982) 481 [SPIRES].
T. Reisz, A power counting theorem for Feynman integrals on the lattice, Comm. Math. Phys. 116 (1988) 81 [SPIRES].
J. Giedt, R. Koniuk, E. Poppitz and T. Yavin, Less naive about supersymmetric lattice quantum mechanics, JHEP 12 (2004) 033 [hep-lat/0410041] [SPIRES].
S. Catterall and E. Gregory, A lattice path integral for supersymmetric quantum mechanics, Phys. Lett. B 487 (2000) 349 [hep-lat/0006013] [SPIRES].
M.F.L. Golterman and D.N. Petcher, A local interactive lattice model with supersymmetry, Nucl. Phys. B 319 (1989) 307 [SPIRES].
L.H. Karsten and J. Smit, The vacuum polarization with SLAC lattice fermions, Phys. Lett. B 85 (1979) 100 [SPIRES].
J.M. Rabin, Perturbation theory for SLAC lattice fermions, Phys. Rev. D 24 (1981) 3218 [SPIRES].
L.H. Karsten and J. Smit, Lattice fermions: species doubling, chiral invariance and the triangle anomaly, Nucl. Phys. B 183 (1981) 103 [SPIRES].
H.S. Sharatchandra, The continuum limit of lattice gauge theories in the context of renormalized perturbation theory, Phys. Rev. D 18 (1978) 2042 [SPIRES].
D. Kadoh and H. Suzuki, Supersymmetric nonperturbative formulation of the WZ model in lower dimensions, arXiv:0909.3686 [SPIRES].
S. Duane, A.D. Kennedy, B.J. Pendleton and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195 (1987) 216 [SPIRES].
K. Fujikawa, Supersymmetry on the lattice and the Leibniz rule, Nucl. Phys. B 636 (2002) 80 [hep-th/0205095] [SPIRES].
S.D. Drell, M. WEinstein and S. Yankielowicz, Variational approach to strong coupling field theory. 1. \( {\phi^4} \) theory, Phys. Rev. D 14 (1976) 487 [SPIRES].
S.D. Drell, M. WEinstein and S. Yankielowicz, Strong coupling field theories. 2. Fermions and gauge fields on a lattice, Phys. Rev. D 14 (1976) 1627 [SPIRES].
A. Kirchberg, J.D. Lange and A. Wipf, From the Dirac operator to Wess-Zumino models on spatial lattices, Ann. Phys. 316 (2005) 357 [hep-th/0407207] [SPIRES].
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ArXiv ePrint: 0909.4791v2
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Bergner, G. Complete supersymmetry on the lattice and a No-Go theorem. J. High Energ. Phys. 2010, 24 (2010). https://doi.org/10.1007/JHEP01(2010)024
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DOI: https://doi.org/10.1007/JHEP01(2010)024