Hierarchy problem, gauge coupling unification at the Planck scale

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
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Realization of vacuum stability
• β-function of Higgs quartic coupling
• Vacuum can be stable by change of gauge coupling running
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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
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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
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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
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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.
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Relation between M* and MGUT
• For fixed b’3 - b’2, relation between M* and MGUT
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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
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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
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Examples of the GCU
• When extra scalars are two SU(2) doublet,
is realized by following extra fermions
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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.
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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
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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
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Backup
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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
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Conditions of the GCU at the Planck scale
• Condition:
• In order to avoid Landau pole (
)
corresponding to
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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)
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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!!
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β-functions
• β-functions up to one-loop
• β-functions of λ up to two-loop
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Boundary conditions
• Boundary conditions
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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
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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
:
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