Hierarchy problem, gauge coupling unification at the Planck scale, and vacuum stability 山口雄也(島根大、北大) 共同研究者:波場直之(島根大)、石田裕之(島根大)、 高橋亮(東北大) 2015/1/7 北大研究発表会 1 Introduction • The SM is completed by the Higgs discovery • suggests the vacuum is meta-stable [Buttazzo, et al., arXiv:1307.3536] • Hierarchy problem: In general, Higgs mass is given quantum corrections by heavy particles as Heavy particle mass 2015/1/7 北大研究発表会 2 Realization of vacuum stability • β-function of Higgs quartic coupling • Vacuum can be stable by change of gauge coupling running 2015/1/7 北大研究発表会 3 Realization of vacuum stability • β-function of Higgs quartic coupling • Vacuum can be stable by change of gauge coupling running 2 3 1 1 Topic: Hierarchy problem + GCU + Vacuum stability ① 2015/1/7 ② 北大研究発表会 ③ 4 Gauge couplings in the SM • RGEs of gauge couplings: • Coefficient bi is calculated by GUT normalization The GCU does not occur ↓ We consider extra particles around the TeV scale 2015/1/7 北大研究発表会 5 Requirement for the GCU • RGE: • The GCU condition where is written by , and is contribution of the extra particles. • Once M* and MGUT are fixed, we can see the necessary values of b’i 2015/1/7 北大研究発表会 6 Extra particles and their contributions to bi Weyl Fermion Complex Scalar *Fermions are included as vector-like for anomaly free, except for real (adjoint) representation. 2015/1/7 北大研究発表会 7 Relation between M* and MGUT • For fixed b’3 - b’2, relation between M* and MGUT 2015/1/7 北大研究発表会 8 Relation between M* and MGUT • For fixed b’3 - b’2, relation between M* and MGUT For M* = 1TeV For only extra fermions (red solid lines) The SM only with TeV scale extra fermions cannot realize the GCU at the Planck scale!! (up to one-loop level) • With two-loop RGEs and one-loop threshold corrections, the GCU could be realized at the Planck scale for 2015/1/7 北大研究発表会 9 Realization of the GCU • With extra fermions and scalars – – *The lower and upper bound of b’3 are given by b’1 ≥ 0 and , respectively. To avoid Landau pole 2015/1/7 北大研究発表会 10 Examples of the GCU • When extra scalars are two SU(2) doublet, is realized by following extra fermions 2015/1/7 北大研究発表会 11 Examples of the GCU 2 • When extra fermions have different masses, is realized by following extra fermions W×1 (0.5) means one (1, 3, 0) fermions with a mass of 0.5TeV. 2015/1/7 北大研究発表会 12 The GCU and vacuum stability • When gauge couplings are large, … smaller larger larger larger smaller larger – The GCU can be realized – The vacuum can be stable 2015/1/7 北大研究発表会 13 Summary • The SM is completed by the Higgs discovery – Higgs mass has quantum corrections as – The vacuum is meta-stable • With extra particles around the TeV scale – The quantum corrections of Higgs mass is O(1)TeV – Runnings of gauge couplings change • The GCU at the Planck scale can be realized • The vacuum can be stable up to the Planck scale • For only extra fermions – With same masses, GCU@Planck cannot be realized (up to one-loop analysis) – With different masses, GCU@Planck can be realized 2015/1/7 北大研究発表会 14 Backup 2015/1/7 北大研究発表会 15 Minimum of λ • Relation between the minimal value of λ and the energy which minimizes λ (by use of two-loop RGEs) Red point • The MPCP could be realized at O(1017)GeV by use of lighter magnitude of top mass as 171GeV 2015/1/7 北大研究発表会 16 Conditions of the GCU at the Planck scale • Condition: • In order to avoid Landau pole ( ) corresponding to 2015/1/7 北大研究発表会 17 The GCU by extra fermions • The smallest value of b’2 and b’3 are 2/3 → • Extra fermions cannot satisfy – – The SM only with TeV scale extra fermions cannot realize the GCU at the Planck scale!! (up to one-loop level) 2015/1/7 北大研究発表会 18 The GCU by extra fermions and scalars • The smallest value of b’2 and b’3 are 1/6 → • Extra particles can satisfy – – The SM with TeV scale extra fermions and scalars can realize the GCU at the Planck scale!! 2015/1/7 北大研究発表会 19 β-functions • β-functions up to one-loop • β-functions of λ up to two-loop 2015/1/7 北大研究発表会 20 Boundary conditions • Boundary conditions 2015/1/7 北大研究発表会 21 The GCU and vacuum stability • When gauge couplings are large, … smaller larger *In general, extra scalar contribution make λ be large. larger larger larger smaller larger – The GCU can be realized – The vacuum can be stable 2015/1/7 北大研究発表会 22 Proton lifetime • Although we do not discuss any specific GUT model, the proton lifetime should be long enough to avoid the experimental lower bound ( τ ~ O(1034) yrs). • The proton lifetime is usually given by a four-fermion approximation for the decay channel : 2015/1/7 北大研究発表会 23
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