Species doublers as super multiplets in lattice supersymmetry: chiral conditions of Wess-Zumino model for \(D = N = 2\). (English) Zbl 1309.81188
Summary: We propose an algebraic lattice supersymmetry formulation which has an exact supersymmetry on the lattice. We show how lattice version of chiral conditions can be imposed to satisfy an exact lattice supersymmetry algebra. The species doublers of chiral fermions and the corresponding bosonic counterparts can be accommodated to fit into chiral supermultiplets of lattice supersymmetry and thus lattice chiral fermion problem does not appear. We explicitly show how \(N = 2\) Wess-Zumino model in one and two dimensions can be formulated to keep exact supersymmetry for all supercharges on the lattice. The momentum representation of \(N = 2\) lattice chiral supersymmetry algebra has lattice periodicity and thus momentum conservation should be modified to a lattice version of sine momentum conservation, which generates nonlocal interactions and leads to a loss of lattice translational invariance. It is shown that the nonlocality is mild and the translational invariance is recovered in the continuum limit. In the coordinate representation a new type of product is defined and the difference operator satisfies Leibnitz rule and an exact lattice supersymmetry is realized on this product.
MSC:
| 81T25 | Quantum field theory on lattices |
| 81T60 | Supersymmetric field theories in quantum mechanics |
References:
| [1] | I. Montvay, Supersymmetric Yang-Mills theory on the lattice, Int. J. Mod. Phys. A 17 (2002) 2377 [hep-lat/0112007] [INSPIRE]. · Zbl 1020.81047 |
| [2] | A. Feo, Supersymmetry on the lattice, Nucl. Phys. Proc. Suppl. 119 (2003) 198 [hep-lat/0210015] [INSPIRE]. · Zbl 1097.81747 · doi:10.1016/S0920-5632(03)01507-X |
| [3] | D.B. Kaplan, Recent developments in lattice supersymmetry, Nucl. Phys. Proc. Suppl. 129 (2004) 109 [hep-lat/0309099] [INSPIRE]. · doi:10.1016/S0920-5632(03)02512-X |
| [4] | S. Catterall, A geometrical approach to N = 2 super Yang-Mills theory on the two-dimensional lattice, JHEP11 (2004) 006 [hep-lat/0410052] [INSPIRE]. · doi:10.1088/1126-6708/2004/11/006 |
| [5] | J. Giedt, Advances and applications of lattice supersymmetry, PoS(LAT2006)008 [hep-lat/0701006] [INSPIRE]. · Zbl 1175.81156 |
| [6] | S. Catterall, D.B. Kaplan and M. Ünsal, Exact lattice supersymmetry, Phys. Rept. 484 (2009) 71 [arXiv:0903.4881] [INSPIRE]. · doi:10.1016/j.physrep.2009.09.001 |
| [7] | P. Dondi and H. Nicolai, Lattice supersymmetry, Nuovo Cim. A 41 (1977) 1 [INSPIRE]. · doi:10.1007/BF02730448 |
| [8] | P. Hasenfratz, Prospects for perfect actions, Nucl. Phys. Proc. Suppl.63 (1998) 53 [hep-lat/9709110] [INSPIRE]. · doi:10.1016/S0920-5632(97)00696-8 |
| [9] | H. Neuberger, More about exactly massless quarks on the lattice, Phys. Lett. B 427 (1998) 353 [hep-lat/9801031] [INSPIRE]. |
| [10] | M. Lüscher, Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation, Phys. Lett. B 428 (1998) 342 [hep-lat/9802011] [INSPIRE]. |
| [11] | K. Fujikawa, Supersymmetry on the lattice and the Leibniz rule, Nucl. Phys. B 636 (2002) 80 [hep-th/0205095] [INSPIRE]. · Zbl 0996.81098 · doi:10.1016/S0550-3213(02)00443-1 |
| [12] | S. Catterall and E. Gregory, A lattice path integral for supersymmetric quantum mechanics, Phys. Lett. B 487 (2000) 349 [hep-lat/0006013] [INSPIRE]. · Zbl 0989.81526 |
| [13] | J. Giedt, R. Koniuk, E. Poppitz and T. Yavin, Less naive about supersymmetric lattice quantum mechanics, JHEP12 (2004) 033 [hep-lat/0410041] [INSPIRE]. · doi:10.1088/1126-6708/2004/12/033 |
| [14] | G. Bergner, Complete supersymmetry on the lattice and a no-go theorem, JHEP01 (2010) 024 [arXiv:0909.4791] [INSPIRE]. · Zbl 1269.81081 · doi:10.1007/JHEP01(2010)024 |
| [15] | 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] [INSPIRE]. |
| [16] | M. Kato, M. Sakamoto and H. So, Taming the Leibniz rule on the lattice, JHEP05 (2008) 057 [arXiv:0803.3121] [INSPIRE]. · doi:10.1088/1126-6708/2008/05/057 |
| [17] | M. Kato, M. Sakamoto and H. So, Leibniz rule and exact supersymmetry on lattice: a case of supersymmetrical quantum mechanics, PoS(LAT2005)274 [hep-lat/0509149] [INSPIRE]. |
| [18] | M. Kato, M. Sakamoto and H. So, No-go theorem of Leibniz rule and supersymmetry on the lattice, PoS(LATTICE 2008)233 [arXiv:0810.2360] [INSPIRE]. |
| [19] | S. Nojiri, Continuous ‘translation’ and supersymmetry on the lattice, Prog. Theor. Phys. 74 (1985) 819 [INSPIRE]. · Zbl 0979.81557 · doi:10.1143/PTP.74.819 |
| [20] | S. Nojiri, The spontaneous breakdown of supersymmetry on the finite lattice, Prog. Theor. Phys. 74 (1985) 1124 [INSPIRE]. · Zbl 0979.81504 · doi:10.1143/PTP.74.1124 |
| [21] | J. Bartels and G. Kramer, A lattice version of the Wess-Zumino model, Z. Phys. C 20 (1983) 159 [INSPIRE]. |
| [22] | A. D’Adda, I. Kanamori, N. Kawamoto and K. Nagata, Twisted superspace on a lattice, Nucl. Phys. B 707 (2005) 100 [hep-lat/0406029] [INSPIRE]. · Zbl 1160.81464 · doi:10.1016/j.nuclphysb.2004.11.046 |
| [23] | A. D’Adda, I. Kanamori, N. Kawamoto and K. Nagata, Exact extended supersymmetry on a lattice: twisted N = 2 super Yang-Mills in two dimensions, Phys. Lett.B 633 (2006) 645 [hep-lat/0507029] [INSPIRE]. · Zbl 1247.81464 |
| [24] | A. D’Adda, I. Kanamori, N. Kawamoto and K. Nagata, Exact extended supersymmetry on a lattice: twisted N = 4 super Yang-Mills in three dimensions, Nucl. Phys.B 798 (2008) 168 [arXiv:0707.3533] [INSPIRE]. · Zbl 1234.81103 · doi:10.1016/j.nuclphysb.2008.01.026 |
| [25] | S. Drell, M. Weinstein and S. Yankielowicz, Strong coupling field theories. 2. Fermions and gauge fields on a lattice, Phys. Rev. D 14 (1976) 1627 [INSPIRE]. |
| [26] | A. D’Adda, A. Feo, I. Kanamori, N. Kawamoto and J. Saito, Species doublers as super multiplets in lattice supersymmetry: exact supersymmetry with interactions for D = 1 N = 2, JHEP09 (2010) 059 [arXiv:1006.2046] [INSPIRE]. · Zbl 1291.81280 · doi:10.1007/JHEP09(2010)059 |
| [27] | L.H. Karsten and J. Smit, The vacuum polarization with SLAC lattice fermions, Phys. Lett.B 85 (1979) 100 [INSPIRE]. |
| [28] | 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] [INSPIRE]. |
| [29] | 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] [INSPIRE]. |
| [30] | F. Bruckmann, S. Catterall and M. de Kok, A critique of the link approach to exact lattice supersymmetry, Phys. Rev. D 75 (2007) 045016 [hep-lat/0611001] [INSPIRE]. |
| [31] | A. D’Adda, N. Kawamoto and J. Saito, Lattice supersymmetry with Hopf algebra for the link approach, PoS(LAT2009)047 [arXiv:0910.3149] [INSPIRE]. |
| [32] | S. Elitzur, E. Rabinovici and A. Schwimmer, Supersymmetric models on the lattice, Phys. Lett. B 119 (1982) 165 [INSPIRE]. |
| [33] | D.B. Kaplan, E. Katz and M. Ünsal, Supersymmetry on a spatial lattice, JHEP05 (2003) 037 [hep-lat/0206019] [INSPIRE]. · doi:10.1088/1126-6708/2003/05/037 |
| [34] | A.G. Cohen, D.B. Kaplan, E. Katz and M. Ünsal, Supersymmetry on a Euclidean space-time lattice. 1. A target theory with four supercharges, JHEP08 (2003) 024 [hep-lat/0302017] [INSPIRE]. · doi:10.1088/1126-6708/2003/08/024 |
| [35] | A.G. Cohen, D.B. Kaplan, E. Katz and M. Ünsal, Supersymmetry on a Euclidean space-time lattice. 2. Target theories with eight supercharges, JHEP12 (2003) 031 [hep-lat/0307012] [INSPIRE]. · doi:10.1088/1126-6708/2003/12/031 |
| [36] | S. Catterall, A geometrical approach to N = 2 super Yang-Mills theory on the two-dimensional lattice, JHEP11 (2004) 006 [hep-lat/0410052] [INSPIRE]. · doi:10.1088/1126-6708/2004/11/006 |
| [37] | F. Sugino, A lattice formulation of super Yang-Mills theories with exact supersymmetry, JHEP01 (2004) 015 [hep-lat/0311021] [INSPIRE]. · Zbl 1243.81125 · doi:10.1088/1126-6708/2004/01/015 |
| [38] | P.H. Damgaard and S. Matsuura, Lattice supersymmetry: equivalence between the link approach and orbifolding, JHEP09 (2007) 097 [arXiv:0708.4129] [INSPIRE]. · doi:10.1088/1126-6708/2007/09/097 |
| [39] | P.H. Damgaard and S. Matsuura, Geometry of orbifolded supersymmetric lattice gauge theories, Phys. Lett. B 661 (2008) 52 [arXiv:0801.2936] [INSPIRE]. · Zbl 1246.81179 |
| [40] | T. Takimi, Relationship between various supersymmetric lattice models, JHEP07 (2007) 010 [arXiv:0705.3831] [INSPIRE]. · doi:10.1088/1126-6708/2007/07/010 |
| [41] | S. Arianos, A. D’Adda, A. Feo, N. Kawamoto and J. Saito, Matrix formulation of superspace on 1D lattice with two supercharges, Int. J. Mod. Phys.A 24 (2009) 4737 [arXiv:0806.0686] [INSPIRE]. · Zbl 1179.81121 |
| [42] | A. D’Adda, A. Feo, I. Kanamori, N. Kawamoto and J. Saito, Lattice supersymmetry: some ideas from low dimensional models, PoS(LAT2009)051 [arXiv:0910.2924] [INSPIRE]. |
| [43] | N. Kawamoto and T. Tsukioka, N = 2 supersymmetric model with Dirac-Kähler fermions from generalized gauge theory in two-dimensions, Phys. Rev.D 61 (2000) 105009 [hep-th/9905222] [INSPIRE]. |
| [44] | J. Kato, N. Kawamoto and Y. Uchida, Twisted superspace for N = D = 2 super BF and Yang-Mills with Dirac-Kähler fermion mechanism, Int. J. Mod. Phys.A 19 (2004) 2149 [hep-th/0310242] [INSPIRE]. · Zbl 1080.81064 |
| [45] | J. Kato, N. Kawamoto and A. Miyake, N = 4 twisted superspace from Dirac-Kähler twist and off-shell SUSY invariant actions in four dimensions, Nucl. Phys. B 721 (2005) 229 [hep-th/0502119] [INSPIRE]. · Zbl 1128.81326 · doi:10.1016/j.nuclphysb.2005.05.024 |
| [46] | H.B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice. 1. Proof by homotopy theory, Nucl. Phys. B 185 (1981) 20 [Erratum ibid.B 195 (1982) 541] [INSPIRE]. · doi:10.1016/0550-3213(81)90361-8 |
| [47] | L.H. Karsten and J. Smit, Lattice fermions: species doubling, chiral invariance and the triangle anomaly, Nucl. Phys. B 183 (1981) 103 [INSPIRE]. · doi:10.1016/0550-3213(81)90549-6 |
| [48] | N. Kawamoto and J. Smit, Effective Lagrangian and dynamical symmetry breaking in strongly coupled lattice QCD, Nucl. Phys. B 192 (1981) 100 [INSPIRE]. · doi:10.1016/0550-3213(81)90196-6 |
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