Abstract
We study the existence of diagonal representatives in each equivalence class of representation matrices of boundary conditions in SU(n) or U(n) gauge theories compactified on the orbifolds T2/ℤN (N = 2, 3, 4, 6). We suppose that the theory has a global G′ = U(n) symmetry. Using constraints, unitary transformations and gauge transformations, we examine whether the representation matrices can simultaneously become diagonal or not. We show that at least one diagonal representative necessarily exists in each equivalence class on T2/ℤ2 and T2/ℤ3, but the representation matrices on T2/ℤ4 and T2/ℤ6 can contain not only diagonal matrices but also non-diagonal 2 × 2 ones and non-diagonal 3 × 3 and 2 × 2 ones, respectively, as members of block-diagonal submatrices. These non-diagonal matrices have discrete parameters, which means that the rank-reducing symmetry breaking can be caused by the discrete Wilson line phases.
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References
N.S. Manton, A New Six-Dimensional Approach to the Weinberg-Salam Model, Nucl. Phys. B 158 (1979) 141 [INSPIRE].
H. Georgi and S.L. Glashow, Unity of All Elementary Particle Forces, Phys. Rev. Lett. 32 (1974) 438 [INSPIRE].
S. Dimopoulos and H. Georgi, Softly Broken Supersymmetry and SU(5), Nucl. Phys. B 193 (1981) 150 [INSPIRE].
N. Sakai, Naturalness in Supersymmetric Guts, Z. Phys. C 11 (1981) 153 [INSPIRE].
Y. Kawamura, Gauge symmetry breaking from extra space S1/Z2, Prog. Theor. Phys. 103 (2000) 613 [hep-ph/9902423] [INSPIRE].
Y. Kawamura, Triplet doublet splitting, proton stability and extra dimension, Prog. Theor. Phys. 105 (2001) 999 [hep-ph/0012125] [INSPIRE].
L.J. Hall and Y. Nomura, Gauge unification in higher dimensions, Phys. Rev. D 64 (2001) 055003 [hep-ph/0103125] [INSPIRE].
M. Kubo, C.S. Lim and H. Yamashita, The Hosotani mechanism in bulk gauge theories with an orbifold extra space S1/Z2, Mod. Phys. Lett. A 17 (2002) 2249 [hep-ph/0111327] [INSPIRE].
C. Csaki, C. Grojean and H. Murayama, Standard model Higgs from higher dimensional gauge fields, Phys. Rev. D 67 (2003) 085012 [hep-ph/0210133] [INSPIRE].
C.A. Scrucca, M. Serone and L. Silvestrini, Electroweak symmetry breaking and fermion masses from extra dimensions, Nucl. Phys. B 669 (2003) 128 [hep-ph/0304220] [INSPIRE].
N.V. Krasnikov, Ultraviolet fixed point behavior of the five-dimensional Yang-Mills theory, the gauge hierarchy problem and a possible new dimension at the TeV scale, Phys. Lett. B 273 (1991) 246 [INSPIRE].
H. Hatanaka, T. Inami and C.S. Lim, The Gauge hierarchy problem and higher dimensional gauge theories, Mod. Phys. Lett. A 13 (1998) 2601 [hep-th/9805067] [INSPIRE].
N. Arkani-Hamed, A.G. Cohen and H. Georgi, Electroweak symmetry breaking from dimensional deconstruction, Phys. Lett. B 513 (2001) 232 [hep-ph/0105239] [INSPIRE].
N. Maru and T. Yamashita, Two-loop Calculation of Higgs Mass in Gauge-Higgs Unification: 5D Massless QED Compactified on S1, Nucl. Phys. B 754 (2006) 127 [hep-ph/0603237] [INSPIRE].
Y. Hosotani, N. Maru, K. Takenaga and T. Yamashita, Two Loop finiteness of Higgs mass and potential in the gauge-Higgs unification, Prog. Theor. Phys. 118 (2007) 1053 [arXiv:0709.2844] [INSPIRE].
J. Hisano, Y. Shoji and A. Yamada, To be, or not to be finite? The Higgs potential in Gauge Higgs Unification, JHEP 02 (2020) 193 [arXiv:1908.09158] [INSPIRE].
N. Haba, Y. Hosotani, Y. Kawamura and T. Yamashita, Dynamical symmetry breaking in gauge Higgs unification on orbifold, Phys. Rev. D 70 (2004) 015010 [hep-ph/0401183] [INSPIRE].
C.S. Lim and N. Maru, Towards a realistic grand gauge-Higgs unification, Phys. Lett. B 653 (2007) 320 [arXiv:0706.1397] [INSPIRE].
Y. Hosotani and N. Yamatsu, Gauge-Higgs grand unification, PTEP 2015 (2015) 111B01 [arXiv:1504.03817] [INSPIRE].
K. Kojima, K. Takenaga and T. Yamashita, The Standard Model Gauge Symmetry from Higher-Rank Unified Groups in Grand Gauge-Higgs Unification Models, JHEP 06 (2017) 018 [arXiv:1704.04840] [INSPIRE].
K. Kojima, K. Takenaga and T. Yamashita, Grand Gauge-Higgs Unification, Phys. Rev. D 84 (2011) 051701 [arXiv:1103.1234] [INSPIRE].
K. Kojima, K. Takenaga and T. Yamashita, Gauge symmetry breaking patterns in an SU(5) grand gauge-Higgs unification model, Phys. Rev. D 95 (2017) 015021 [arXiv:1608.05496] [INSPIRE].
T. Yamashita, Doublet-Triplet Splitting in an SU(5) Grand Unification, Phys. Rev. D 84 (2011) 115016 [arXiv:1106.3229] [INSPIRE].
M. Kakizaki, S. Kanemura, H. Taniguchi and T. Yamashita, Higgs sector as a probe of supersymmetric grand unification with the Hosotani mechanism, Phys. Rev. D 89 (2014) 075013 [arXiv:1312.7575] [INSPIRE].
H. Nakano, M. Sato, O. Seto and T. Yamashita, Dirac gaugino from grand gauge-Higgs unification, PTEP 2022 (2022) 033B06 [arXiv:2201.04428] [INSPIRE].
Y. Hosotani, Dynamical Mass Generation by Compact Extra Dimensions, Phys. Lett. B 126 (1983) 309 [INSPIRE].
Y. Hosotani, Dynamics of Nonintegrable Phases and Gauge Symmetry Breaking, Annals Phys. 190 (1989) 233 [INSPIRE].
N. Haba, M. Harada, Y. Hosotani and Y. Kawamura, Dynamical rearrangement of gauge symmetry on the orbifold S1/Z2, Nucl. Phys. B 657 (2003) 169 [hep-ph/0212035] [INSPIRE].
N. Haba, Y. Hosotani and Y. Kawamura, Classification and dynamics of equivalence classes in SU(N) gauge theory on the orbifold S1/Z2, Prog. Theor. Phys. 111 (2004) 265 [hep-ph/0309088] [INSPIRE].
N. Haba and T. Yamashita, A General formula of the effective potential in 5-D SU(N) gauge theory on orbifold, JHEP 02 (2004) 059 [hep-ph/0401185] [INSPIRE].
Y. Kawamura, T. Kinami and T. Miura, Equivalence Classes of Boundary Conditions in Gauge Theory on Z3 Orbifold, Prog. Theor. Phys. 120 (2008) 815 [arXiv:0808.2333] [INSPIRE].
Y. Kawamura and T. Miura, Equivalence Classes of Boundary Conditions in SU(N) Gauge Theory on 2-dimensional Orbifolds, Prog. Theor. Phys. 122 (2010) 847 [arXiv:0905.4123] [INSPIRE].
Y. Goto and Y. Kawamura, Orbifold family unification using vectorlike representation on six dimensions, Phys. Rev. D 98 (2018) 035039 [arXiv:1712.06444] [INSPIRE].
Y. Hosotani, S. Noda and K. Takenaga, Dynamical gauge symmetry breaking and mass generation on the orbifold T2/Z2, Phys. Rev. D 69 (2004) 125014 [hep-ph/0403106] [INSPIRE].
Y. Kawamura and Y. Nishikawa, On diagonal representatives in boundary condition matrices on orbifolds, Int. J. Mod. Phys. A 35 (2020) 2050206 [arXiv:2009.10958] [INSPIRE].
G. ’t Hooft, A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories, Nucl. Phys. B 153 (1979) 141 [INSPIRE].
G. von Gersdorff, A New Class of Rank Breaking Orbifolds, Nucl. Phys. B 793 (2008) 192 [arXiv:0705.2410] [INSPIRE].
C. Bachas, A Way to break supersymmetry, hep-th/9503030 [INSPIRE].
C.A. Scrucca and M. Serone, Anomalies in field theories with extra dimensions, Int. J. Mod. Phys. A 19 (2004) 2579 [hep-th/0403163] [INSPIRE].
C.A. Scrucca, M. Serone, L. Silvestrini and A. Wulzer, Gauge Higgs unification in orbifold models, JHEP 02 (2004) 049 [hep-th/0312267] [INSPIRE].
S. Förste, H.P. Nilles and A. Wingerter, Geometry of rank reduction, Phys. Rev. D 72 (2005) 026001 [hep-th/0504117] [INSPIRE].
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Kawamura, Y., Kodaira, E., Kojima, K. et al. On representation matrices of boundary conditions in SU(n) gauge theories compactified on two-dimensional orbifolds. J. High Energ. Phys. 2023, 113 (2023). https://doi.org/10.1007/JHEP04(2023)113
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DOI: https://doi.org/10.1007/JHEP04(2023)113