To be, or not to be finite? The Higgs potential in Gauge-Higgs Unification
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Springer
Received: September 3, Revised: November 27, Accepted: December 19, Published: February 28,
2019 2019 2019 2020
Junji Hisano,a,b,c Yutaro Shojia and Atsuyuki Yamadab a
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Furo-cho Chikusa-ku, Nagoya, 464-8602 Japan b Department of Physics, Nagoya University, Furo-cho Chikusa-ku, Nagoya, 464-8602 Japan c Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, Chiba, 277-8584 Japan
E-mail: [email protected], [email protected], [email protected] Abstract: In this paper, we investigate the finiteness of the Higgs effective potential in an SU(N ) Gauge-Higgs Unification (GHU) model defined on M4 × S 1 . We obtain the Higgs effective potential at the two-loop level and find that it is finite. We also discuss that the Higgs effective potential is generically divergent for three- or higher-loop levels. As an example, we consider an SU(N ) gauge theory on M5 ×S 1 , where the one-loop corrections to the four-Fermi operators are divergent. We find that the Higgs effective potential depends on their counter terms at the three-loop level. Keywords: Phenomenology of Field Theories in Higher Dimensions ArXiv ePrint: 1908.09158
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP02(2020)193
JHEP02(2020)193
To be, or not to be finite? The Higgs potential in Gauge-Higgs Unification
Contents 1
2 Dynamical symmetry breaking by Hosotani mechanism
3
3 Compactification by superposition
4
4 Higgs effective potential up to two-loop level 4.1 One-loop effective potential 4.2 Two-loop effective potential
6 6 7
5 Divergences at higher-loop level
9
6 Summary
10
A Proof of identities A.1 Proof of eq. (3.5) A.2 Proof of eq. (4.6) A.3 Symmetries of Gadj and G`
11 11 12 12
B Momentum integrals B.1 Momentum integrals with a spacial shift operator B.2 Proof of F (0) = 0
13 13 14
C Example of the two-loop calculation
16
1
Introduction
The Higgs mechanism is one of the essential ingredients in the standard model (SM) of particle physics. It generates masses for the gauge bosons and the fermions, which were forbidden by the gauge symmetries of the standard model. Consequently, all the masses are described by the Higgs vacuum expectation value (VEV) and the couplings, which is now in good agreement with the Higgs coupling measurements at Large Hadron Collider [1, 2]. In spite of the importance of the mechanism, the nature of the Higgs boson has not been understood well. It has been discussed for a long time that a scalar field is very sensitive to a UV cutoff scale, such as the Planck scale or the grand unification scale, and it is not natural that the Higgs VEV lies around the electroweak (EW) scale. If the Higgs boson is really a fundamental scalar field, one needs to protect the Higgs mass term from dangerous quantum corrections, which is greatly achieved by supersymmetry [3–7]. Alternatively, one can assume that the Higgs boson originates from fields with
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