Minimal Scale Invariant Theory of Electroweak Symmetry ......16π2 (λ−3h 2 t +···) Λ2 If Λ...
Transcript of Minimal Scale Invariant Theory of Electroweak Symmetry ......16π2 (λ−3h 2 t +···) Λ2 If Λ...
-
Minimal Scale Invariant Theory of
Electroweak Symmetry Breaking
Apostolos Pilaftsis
School of Physics and Astronomy, The University of Manchester,
Manchester M13 9PL, United Kingdom
&
Department of Theoretical Physics and IFIC, University of Valencia,
E-46100, Valencia, Spain
CERN, 16 June 2011
Based on JHEP09 (2010) 021 (63 pages), with Lisa Alexander–Nunneley
-
Plan of the talk
• The Standard Theory of Electroweak Symmetry Breaking: SM
• The Gauge Hierarchy Problem and Proposed Solutions
• Classical Scale Symmetry and Flat Directions
• The Minimal Scale Invariant extension of the Standard Model:MSISM
• Phenomenology
• Conclusions
CERN, 16 June 2011 A. Pilaftsis
-
• The Standard Theory of Electroweak Symmetry Breaking
Higgs Mechanism in the SM: SU(3)colour⊗ SU(2)L⊗U(1)Y
Φ
V (Φ)
〈Φ
〉
Higgs potential V (Φ)
V (Φ) = −m2Φ†Φ + λ(Φ†Φ)2 .Ground state: 〈
Φ〉
=
√m2
2λ
(01
)
carries weak charge, but no electriccharge and colour.
After Spontaneous Symmetry Breaking:
⇒ W±, Z bosons and matter feel the presence of〈Φ
〉and become massive,
but not γ and ga, e.g. MW = gw〈Φ
〉
⇒ Quantum excitations of Φ =〈Φ
〉+ H
(01
); H is the Higgs boson.
CERN, 16 June 2011 A. Pilaftsis
-
Light SM Higgs boson experimentally favourable
0
1
2
3
4
5
6
10030 300
mH [GeV]
∆χ2
Excluded Preliminary
∆αhad =∆α(5)
0.02758±0.000350.02749±0.00012incl. low Q2 data
Theory uncertaintyJuly 2010 mLimit = 158 GeV
80.3
80.4
80.5
150 175 200
mH [GeV]114 300 1000
mt [GeV]
mW
[G
eV
]
68% CL
∆α
LEP1 and SLD
LEP2 and Tevatron (prel.)
July 2010
[ LEP–TEVATRON EWG, http://lepewwg.web.cern.ch/LEPEWWG/]
CERN, 16 June 2011 A. Pilaftsis
-
• The Gauge Hierarchy Problem
Quantum effects on the Higgs-boson mass MH :
H
+
t
t
· · · : 116π2
(λ − 3h2t + · · · ) Λ2
CERN, 16 June 2011 A. Pilaftsis
-
• The Gauge Hierarchy Problem
Quantum effects on the Higgs-boson mass MH :
H
+
t
t
· · · : 116π2
(λ − 3h2t + · · · ) Λ2
If Λ ∼ MGUT, Higgs-mass CT δM2H ∼ 116π2M2GUT → δM2H ∼ 1024M2H!
CERN, 16 June 2011 A. Pilaftsis
-
• The Gauge Hierarchy Problem
Quantum effects on the Higgs-boson mass MH :
H
+
t
t
· · · : 116π2
(λ − 3h2t + · · · ) Λ2
If Λ ∼ MGUT, Higgs-mass CT δM2H ∼ 116π2M2GUT → δM2H ∼ 1024M2H!
=⇒ Vector bosons are protected from quadratic divergences, if gauged.
CERN, 16 June 2011 A. Pilaftsis
-
• The Gauge Hierarchy Problem
Quantum effects on the Higgs-boson mass MH :
H
+
t
t
· · · : 116π2
(λ − 3h2t + · · · ) Λ2
If Λ ∼ MGUT, Higgs-mass CT δM2H ∼ 116π2M2GUT → δM2H ∼ 1024M2H!
=⇒ Vector bosons are protected from quadratic divergences, if gauged.
=⇒ Fermion masses have no linear divergences because of chirality.
CERN, 16 June 2011 A. Pilaftsis
-
• The Gauge Hierarchy Problem
Quantum effects on the Higgs-boson mass MH :
H
+
t
t
· · · : 116π2
(λ − 3h2t + · · · ) Λ2
If Λ ∼ MGUT, Higgs-mass CT δM2H ∼ 116π2M2GUT → δM2H ∼ 1024M2H!
=⇒ Vector bosons are protected from quadratic divergences, if gauged.
=⇒ Fermion masses have no linear divergences because of chirality.
=⇒ No logarithmic stability of MH against QM effects from MZ toMU ∼ 1016 GeV or MPl ∼ 1019 GeV.
→ This is the Gauge Hierarchy Problem
CERN, 16 June 2011 A. Pilaftsis
-
• Proposed Solutions to the Gauge Hierarchy Problem
→ SUperSYmmetry
→ Flat or Warped Large Extra Dimensions
→ Higgs as a Pseudo-Goldstone Boson
→ No Higgs at All: Technicolour theories, Composite Higgs . . .
→...
CERN, 16 June 2011 A. Pilaftsis
-
• Proposed Solutions to the Gauge Hierarchy Problem
→ SUperSYmmetry
→ Flat or Warped Large Extra Dimensions
→ Higgs as a Pseudo-Goldstone Boson
→ No Higgs at All: Technicolour theories, Composite Higgs . . .
→...
→ Scale or Conformal Symmetry
CERN, 16 June 2011 A. Pilaftsis
-
References
[1] S. R. Coleman and E. Weinberg, “Radiative Corrections As The Origin Of Spontaneous
Symmetry Breaking,” Phys. Rev. D 7 (1973) 1888.
[2] E. Gildener and S. Weinberg, “Symmetry Breaking And Scalar Bosons,”
Phys. Rev. D 13 (1976) 3333.
[3] R. Hempfling, “The Next-to-minimal Coleman-Weinberg model,”
Phys. Lett. B 379 (1996) 153.
[4] K. A. Meissner and H. Nicolai, “Conformal symmetry and the standard model,” Phys.
Lett. B 648 (2007) 312; Phys. Lett. B 660 (2008) 260.
[5] R. Foot, A. Kobakhidze and R. R. Volkas, “Electroweak Higgs as a pseudo-Goldstone
boson of broken scale invariance,” Phys. Lett. B 655 (2007) 156.
[6] W. F. Chang, J. N. Ng and J. M. S. Wu, “Shadow Higgs from a scale-invariant
hidden U(1)s model,” Phys. Rev. D 75 (2007) 115016.
[7] M. Shaposhnikov and D. Zenhäusern, Phys. Lett. B 671 (2009) 162;
D. Blas, M. Shaposhnikov and D. Zenhäusern, “Scale-invariant alternatives to General
Relativity,” arXiv:1104.1329.
[8] L. Alexander-Nunneley and A. Pilaftsis, “The Minimal Scale Invariant Extension of
the Standard Model,” JHEP 1009 (2010) 021.
CERN, 16 June 2011 A. Pilaftsis
-
• Classical Scale Symmetry and Flat Directions
Consider the action of a generic theory:
S[Φ(x)] =
∫d4x L[ ∂µΦ(x), Φ(x)] ,
with
L = 12∂µΦ(x)∂
µΦ(x) +1
2m2Φ2(x) − λΦ4(x) + C .
CERN, 16 June 2011 A. Pilaftsis
-
• Classical Scale Symmetry and Flat Directions
Consider the action of a generic theory:
S[Φ(x)] =
∫d4x L[ ∂µΦ(x), Φ(x)] ,
with
L = 12∂µΦ(x)∂
µΦ(x) +1
2m2Φ2(x) − λΦ4(x) + C .
• Scaling transformation acting on the fields:
Φ(x) → Φ′(x′) = eǫa Φ(eǫx) ,
with a = 1(32
)for bosons (fermions).
Classical Scale Invariance:
S[Φ(x)] = S[Φ′(x′)] =⇒ m2 = 0 , C = 0 .=⇒ No mass-scale in the theory.
CERN, 16 June 2011 A. Pilaftsis
-
• Tree-level Ward identity from scaling symmetry:
If Φ = (φ1, φ2, . . . , φn) = ϕN, with |N| = 1, then
Φ · ∇V tree(Φ) = ϕ ddϕ
V tree(ϕN) = 4V tree(Φ) ,
with ∇ ≡(
∂∂φ1
, ∂∂φ2
, · · · , ∂∂φn
).
CERN, 16 June 2011 A. Pilaftsis
-
• Tree-level Ward identity from scaling symmetry:
If Φ = (φ1, φ2, . . . , φn) = ϕN, with |N| = 1, then
Φ · ∇V tree(Φ) = ϕ ddϕ
V tree(ϕN) = 4V tree(Φ) ,
with ∇ ≡(
∂∂φ1
, ∂∂φ2
, · · · , ∂∂φn
).
• Flat direction along a given unit vector N = n (ϕ 6= 0):
dV tree(ϕn)
dϕ= 0 ⇐⇒ V tree(ϕn) = 0 , for SI theory only.
CERN, 16 June 2011 A. Pilaftsis
-
• Tree-level Ward identity from scaling symmetry:
If Φ = (φ1, φ2, . . . , φn) = ϕN, with |N| = 1, then
Φ · ∇V tree(Φ) = ϕ ddϕ
V tree(ϕN) = 4V tree(Φ) ,
with ∇ ≡(
∂∂φ1
, ∂∂φ2
, · · · , ∂∂φn
).
• Flat direction along a given unit vector N = n (ϕ 6= 0):
dV tree(ϕn)
dϕ= 0 ⇐⇒ V tree(ϕn) = 0 , for SI theory only.
• Condition for an extremal or stationary flat direction:
∇V tree(Φ)∣∣∣Φ=ϕn
= 0 . But, ϕ remains undetermined.
CERN, 16 June 2011 A. Pilaftsis
-
• Tree-level Ward identity from scaling symmetry:
If Φ = (φ1, φ2, . . . , φn) = ϕN, with |N| = 1, then
Φ · ∇V tree(Φ) = ϕ ddϕ
V tree(ϕN) = 4V tree(Φ) ,
with ∇ ≡(
∂∂φ1
, ∂∂φ2
, · · · , ∂∂φn
).
• Flat direction along a given unit vector N = n (ϕ 6= 0):
dV tree(ϕn)
dϕ= 0 ⇐⇒ V tree(ϕn) = 0 , for SI theory only.
• Condition for an extremal or stationary flat direction:
∇V tree(Φ)∣∣∣Φ=ϕn
= 0 . But, ϕ remains undetermined.
• Minimization and convexity conditions:
(v·∇)2V tree(Φ)∣∣∣Φ=ϕn
≥ 0 , V tree(ϕN) ≥ 0 , for all possible v and N.
CERN, 16 June 2011 A. Pilaftsis
-
• Radiative EW breaking of SI: the Gildener–Weinberg approach
General MS-renormalized SI scalar potential:
V tree(Φ) =1
4!fijkl(µ) φiφjφkφl =
ϕ4
4!fijkl(µ) NiNjNkNl ,
with V tree(Φflat = ϕn) = 0, at the RG scale µ = Λ.
CERN, 16 June 2011 A. Pilaftsis
-
• Radiative EW breaking of SI: the Gildener–Weinberg approach
General MS-renormalized SI scalar potential:
V tree(Φ) =1
4!fijkl(µ) φiφjφkφl =
ϕ4
4!fijkl(µ) NiNjNkNl ,
with V tree(Φflat = ϕn) = 0, at the RG scale µ = Λ.
Stationarity condition for a local extremum:
∇(V tree(Φ) + V 1−loopeff (Φ)
)∣∣∣Φ = vϕn + δΦ
= 0 ,
with δΦ ⊥ vϕn and δΦ = O(1-loop).
CERN, 16 June 2011 A. Pilaftsis
-
• Radiative EW breaking of SI: the Gildener–Weinberg approach
General MS-renormalized SI scalar potential:
V tree(Φ) =1
4!fijkl(µ) φiφjφkφl =
ϕ4
4!fijkl(µ) NiNjNkNl ,
with V tree(Φflat = ϕn) = 0, at the RG scale µ = Λ.
Stationarity condition for a local extremum:
∇(V tree(Φ) + V 1−loopeff (Φ)
)∣∣∣Φ = vϕn + δΦ
= 0 ,
with δΦ ⊥ vϕn and δΦ = O(1-loop).
Multiply from left with Φ and use the tree-level WI:
4 V tree(vϕn + δΦ) + Φ · ∇V 1−loopeff (Φ)∣∣∣Φ = vϕn + δΦ
= 0 .
CERN, 16 June 2011 A. Pilaftsis
-
GW perturbative approach to find vϕ:
(4 δΦ · ∇V tree(Φ) + Φ · ∇V 1−loopeff (Φ)
)∣∣∣Φ = vϕn
= 0 .
CERN, 16 June 2011 A. Pilaftsis
-
GW perturbative approach to find vϕ:
(4 δΦ · ∇V tree(Φ) + Φ · ∇V 1−loopeff (Φ)
)∣∣∣Φ = vϕn
= 0 .
Since ∇V tree(Φ)|Φ=vϕn = 0, the extremum along Φflat = ϕn is
n · ∇V 1−loopeff (Φ)∣∣∣Φ = vϕn
=dV
1−loopeff (ϕn)
dϕ
∣∣∣∣∣ϕ = vϕ
= 0 .
CERN, 16 June 2011 A. Pilaftsis
-
GW perturbative approach to find vϕ:
(4 δΦ · ∇V tree(Φ) + Φ · ∇V 1−loopeff (Φ)
)∣∣∣Φ = vϕn
= 0 .
Since ∇V tree(Φ)|Φ=vϕn = 0, the extremum along Φflat = ϕn is
n · ∇V 1−loopeff (Φ)∣∣∣Φ = vϕn
=dV
1−loopeff (ϕn)
dϕ
∣∣∣∣∣ϕ = vϕ
= 0 .
The 1-loop effective potential along Φflat = ϕn is
V1−loopeff (ϕn) = A(n)ϕ
4 + B(n) ϕ4 lnϕ2
Λ2= B(n) ϕ4
(ln
ϕ2
v2ϕ− 1
2
),
where A and B are dimensionless constants (depending on particle masses)and
Λ = vϕ exp
(A
2B+
1
4
).
CERN, 16 June 2011 A. Pilaftsis
-
• Remarks:
–Scalar mass spectrum:
Pseudo-Goldstone boson h associated with the radiative EW breaking of SI:
m2h =d2V
1−loopeff (ϕn)
dϕ2
∣∣∣∣∣ϕ=vϕ
= 8Bv2ϕ .
CERN, 16 June 2011 A. Pilaftsis
-
• Remarks:
–Scalar mass spectrum:
Pseudo-Goldstone boson h associated with the radiative EW breaking of SI:
m2h =d2V
1−loopeff (ϕn)
dϕ2
∣∣∣∣∣ϕ=vϕ
= 8Bv2ϕ .
–SI UV regularization schemes:
UV regularization schemes should preserve the SI WIs of the classical action,such as the dimensional regularization scheme.
=⇒ No need of explicit Mass Counterterm.
CERN, 16 June 2011 A. Pilaftsis
-
• Remarks:
–Scalar mass spectrum:
Pseudo-Goldstone boson h associated with the radiative EW breaking of SI:
m2h =d2V
1−loopeff (ϕn)
dϕ2
∣∣∣∣∣ϕ=vϕ
= 8Bv2ϕ .
–SI UV regularization schemes:
UV regularization schemes should preserve the SI WIs of the classical action,such as the dimensional regularization scheme.
=⇒ No need of explicit Mass Counterterm.
–The Gauge Hierarchy Problem:
Ignoring Quantum Gravity effects, m2h is stable against QM effects.
CERN, 16 June 2011 A. Pilaftsis
-
• The MSISM
• Scalar fields:
Φ =
(G+
1√2(φ + iG)
), S =
1√2
(σ + iJ) = |S| eiθS .
CERN, 16 June 2011 A. Pilaftsis
-
• The MSISM
• Scalar fields:
Φ =
(G+
1√2(φ + iG)
), S =
1√2
(σ + iJ) = |S| eiθS .
• Classical Potential of the MSISM
V tree(Φ, S) =λ1
2(Φ†Φ)2 +
λ2
2(S∗S)2 + λ3 Φ
†Φ S∗S + λ4 Φ†Φ S2
+ λ∗4 Φ†Φ S∗2 + λ5 S
3S∗ + λ∗5 SS∗3 +
λ6
2S4 +
λ∗62
S∗4
=1
2
(Φ†Φ , S∗S
)(λ1 λ3 + 2Re (λ4e
2iθS)
λ3 + 2Re (λ4e2iθS) λ2 + 4Re (λ5e
2iθS) + 2Re (λ6e4iθS)
)
︸ ︷︷ ︸≡ Λ
(Φ†ΦS∗S
).
CERN, 16 June 2011 A. Pilaftsis
-
• Convexity conditions on V tree(Φ, S):
(i) TrΛ ≥ 0 , (ii){
Λ12 ≥ 0 , if Λ11 = 0 or Λ22 = 0DetΛ ≥ 0 , if Λ11 6= 0 and Λ22 6= 0
.
CERN, 16 June 2011 A. Pilaftsis
-
• Convexity conditions on V tree(Φ, S):
(i) TrΛ ≥ 0 , (ii){
Λ12 ≥ 0 , if Λ11 = 0 or Λ22 = 0DetΛ ≥ 0 , if Λ11 6= 0 and Λ22 6= 0
.
• Classification of flat directions:
Φflat = ϕ
nφnσnJ
=
φ
σ
J
; n2φ + n2σ + n2J = 1 .
CERN, 16 June 2011 A. Pilaftsis
-
• Convexity conditions on V tree(Φ, S):
(i) TrΛ ≥ 0 , (ii){
Λ12 ≥ 0 , if Λ11 = 0 or Λ22 = 0DetΛ ≥ 0 , if Λ11 6= 0 and Λ22 6= 0
.
• Classification of flat directions:
Φflat = ϕ
nφnσnJ
=
φ
σ
J
; n2φ + n2σ + n2J = 1 .
–Type-I flat direction: |S| = 0; Φflat = φ =⇒ λ1(Λ) = 0.
–Type-II flat direction: Φflat general, constrained by V tree minimization.
–Type-III flat direction: φ = 0; Φflat = ϕ
(nσnJ
)=⇒ λ2,5,6(Λ) = 0.
CERN, 16 June 2011 A. Pilaftsis
-
• Convexity conditions on V tree(Φ, S):
(i) TrΛ ≥ 0 , (ii){
Λ12 ≥ 0 , if Λ11 = 0 or Λ22 = 0DetΛ ≥ 0 , if Λ11 6= 0 and Λ22 6= 0
.
• Classification of flat directions:
Φflat = ϕ
nφnσnJ
=
φ
σ
J
; n2φ + n2σ + n2J = 1 .
–Type-I flat direction: |S| = 0; Φflat = φ =⇒ λ1(Λ) = 0.
–Type-II flat direction: Φflat general, constrained by V tree minimization.
–Type-III flat direction: φ = 0; Φflat = ϕ
(nσnJ
)=⇒ λ2,5,6(Λ) = 0.
(introduces little hierarchy between vSM and Λ)
CERN, 16 June 2011 A. Pilaftsis
-
• 1-loop effective potential for Type-I & Type-II MSISM:
V1−loopeff (φ) = α φ
4 + β φ4 lnφ2
Λ2= β φ4
(ln
φ2
v2SM− 1
2
),
with vSM ≈ 246 GeV and
β =1
64π2v4SM
(2∑
i=1
m4Hi + 6m4W + 3m
4Z − 12m4t − 2
3∑
i=1
m4Ni
),
Λ = vSM exp
(α
2β+
1
4
), for α ∼ β ⇒ Λ ∼ vSM .
m2h = 8β n2φv
2SM ⇐= Pseudo-Goldstone h-Boson
CERN, 16 June 2011 A. Pilaftsis
-
• 1-loop effective potential for Type-I & Type-II MSISM:
V1−loopeff (φ) = α φ
4 + β φ4 lnφ2
Λ2= β φ4
(ln
φ2
v2SM− 1
2
),
with vSM ≈ 246 GeV and
β =1
64π2v4SM
(2∑
i=1
m4Hi + 6m4W + 3m
4Z − 12m4t − 2
3∑
i=1
m4Ni
),
Λ = vSM exp
(α
2β+
1
4
), for α ∼ β ⇒ Λ ∼ vSM .
m2h = 8β n2φv
2SM ⇐= Pseudo-Goldstone h-Boson
• Important remark on SI SM:
β1−loop =1
64π2v4SM
(6m4W + 3m
4Z − 12m4t
)< 0 .
=⇒ Perturbative SI SM unrealistic! What about non-perturbative SI SM?
CERN, 16 June 2011 A. Pilaftsis
-
• Model Taxonomy
Flat direction U(1) Invariant CP Violation Massive DM SeesawCandidate Neutrinos
Type-I
S = 0 Yes None Yes No
S = 0 No Explicit Yes No
Type-II
S = real Yes None No Yes
S = real No Explicit Model YesDependent
S = imaginary No Explicit Model YesDependent
S = complex No Explicit or Model YesSpontaneous Dependent
CERN, 16 June 2011 A. Pilaftsis
-
• Phenomenology
• EW oblique parameters S, T and U
µ ν
φ
G/G±p
µ ν
φ
Z/W±p µ ν
φ
p
Theoretical prediction: δP = PMSISM − PSM, with P = S, T, U .
Experimental limits:
−0.296 < δSexp < 0.096 ,−0.296 < δTexp < 0.136 ,−0.066 < δUexp < 0.366 ,
for mrefHSM = 117 GeV.
CERN, 16 June 2011 A. Pilaftsis
-
• Perturbativity constraints on couplings
Constraint on 1-loop β-functions:
βλ ≤ 1 ,
where λ ∈ {λ1,2,...,6, g′, g, gs, he,u,d, hN , h̃N}.
CERN, 16 June 2011 A. Pilaftsis
-
• Perturbativity constraints on couplings
Constraint on 1-loop β-functions:
βλ ≤ 1 ,
where λ ∈ {λ1,2,...,6, g′, g, gs, he,u,d, hN , h̃N}.
1-loop RGEs for λ1,2,...,6 and hN , h̃N are calculated, using the
Displacement Operator Formalism for Renormalization.[D. Binosi, J. Papavassiliou and AP, PRD71 (2005) 085007.]
ΓR(φR, λR) = limε→0
eD Γ0(φR, λR ; ε) ,
D = δφ∂
∂φR+ δλ
∂
∂λR,
with δφ = φ0 − φR and δλ = λ0 − λR.
CERN, 16 June 2011 A. Pilaftsis
-
Type-I U(1)-Invariant MSISM
V tree(Λ) =λ2(Λ)
2(S∗S)2 + λ3(Λ) Φ
†Φ S∗S .
Scalar boson mass spectrum:
h ≡ φ , H1 ≡ σ , H2 ≡ J ,
with
m2σ = m2J =
λ3(Λ)
2v2SM .
Higgs-to-gauge boson couplings:
g2hZZ = 1 , g2H1,2ZZ
= 0 .
CERN, 16 June 2011 A. Pilaftsis
-
Type-I U(1)-Invariant MSISM [L. Alexander–Nunneley, AP, JHEP09 (2010) 021]
1 10 10210
102
103
104
Λ3HLL
mhHG
eVL
LEP
∆T
∆S
ΒΛ31
1 10 102102
103
104
Λ3HLL
mΣ
,JHG
eVL
LEP
∆T
∆S
ΒΛ31
CERN, 16 June 2011 A. Pilaftsis
-
General Type-I MSISM
V tree(Λ) =λ2(Λ)
2(S∗S)2 + λ3(Λ)Φ
†Φ S∗S + λ4(Λ) Φ†Φ S2 + λ∗4(Λ) Φ
†Φ S∗2
+ λ5(Λ) S3S∗ + λ∗5(Λ) SS
∗3 +λ6(Λ)
2S4 +
λ∗6(Λ)
2S∗4 .
Scalar boson mass spectrum:
h ≡ φ , H1 = cos θ σ + sin θ J , H2 = − sin θ σ + cos θ J ,
with
m2H1 =1
2
(λ3(Λ)+2|λ4(Λ)|
)v2SM , m
2H2
=1
2
(λ3(Λ)−2|λ4(Λ)|
)v2SM .
Higgs-to-gauge boson couplings:
g2hZZ = 1 , g2H1,2ZZ
= 0 .
CERN, 16 June 2011 A. Pilaftsis
-
General Type-I MSISM [L. Alexander–Nunneley, AP, JHEP09 (2010) 021]
1 10 10210
102
103
104
Λ3HLL
mhHG
eVL
LEP
∆T∆S
ΒΛ31
Λ4=0
Λ3=2 Λ4
1 10 102102
103
104
Λ3HLL
mH
2HG
eVL
LEP
∆T∆S
ΒΛ31
Λ4 =0
Β=0
1 10 102102
103
104
Λ3HLL
mH
1HG
eVL
LEP
∆T∆S
ΒΛ31
Β=0Λ4 =
0
Λ3=2 Λ4
1 10 102102
103
104
Λ3HLL
LHG
eVL
LEP
∆T∆S
ΒΛ31Λ3=2
Λ4
Λ4 =0
CERN, 16 June 2011 A. Pilaftsis
-
Type-II U(1)-Invariant MSISM
V tree(Λ) =λ1(Λ)
2(Φ†Φ)2 +
λ2(Λ)
2(S∗S)2 + λ3(Λ) Φ
†Φ S∗S ,
withφ2
σ2=
n2φ
n2σ= − λ2(Λ)
λ3(Λ)= − λ3(Λ)
λ1(Λ).
Scalar mass spectrum:
h = cos θ φ + sin θ σ , H1 ≡ H = − sin θ φ + cos θ σ , H2 ≡ J ,
withm2H =
[λ1(Λ) − λ3(Λ)
]v2SM , m
2J = 0 .
Higgs-to-gauge boson couplings:
g2hV V = cos2 θ =
−λ3(Λ)λ1(Λ) − λ3(Λ)
, g2HV V = sin2 θ =
λ1(Λ)
λ1(Λ) − λ3(Λ).
CERN, 16 June 2011 A. Pilaftsis
-
Type-II U(1)-Invariant MSISM [L. Alexander–Nunneley, AP, JHEP09 (2010) 021]
0
-1
-2
-3
-4
0 1 2 3 4
Λ3HLL
Λ1HLL
LEP
Β=0
ΒΛ2=1
ΒΛ1=1
Pert0
-0.01
-0.02
-0.03
-0.04
0 1 2 3 4
Λ3HLL
Λ1HLL
LEP
Β=0 ΒΛ1=1
Pert
0 1 2 3 40
50
100
150
Λ1HLL
mhHG
eVL LEP
LEP
ΒΛ2=1 ΒΛ1=1
0 1 2 3 4200
300
400
500
600
700
Λ1HLL
mHHG
eVL
LEP
LEP
ΒΛ2=1ΒΛ1=1
Λ3=0Β=0
CERN, 16 June 2011 A. Pilaftsis
-
Type-II MSISM with Maximal SCPV[L. Alexander–Nunneley, AP, JHEP09 (2010) 021]
Breaking pattern U(1) → Z4 symmetry:
V tree(Λ) =λ1(Λ)
2(Φ†Φ)2+
λ2(Λ)
2(S∗S)2+λ3(Λ) Φ
†Φ S∗S+λ6(Λ)
2(S4+S∗4) ,
with
φ2
σ2=
n2φ
n2σ= −2λ3(Λ)
λ1(Λ)= −2
[λ2(Λ) − 2λ6(Λ)
]
λ3(Λ), σ = J , nσ = nJ .
Scalar boson mass spectrum:
m2H1 =[λ1(Λ)−λ3(Λ)
]v2SM , m
2H2
= 4λ1(Λ)λ6(Λ)
−λ3(Λ)v2SM , H2 =
1√2
(−σ+J) .
Higgs-to-gauge boson couplings:
g2hV V =−λ3(Λ)
λ1(Λ) − λ3(Λ), g2H1V V =
λ1(Λ)
λ1(Λ) − λ3(Λ).
CERN, 16 June 2011 A. Pilaftsis
-
Type-II MSISM with Maximal SCPV[L. Alexander–Nunneley, AP, JHEP09 (2010) 021]
0-0.005-0.01-0.015-0.02-0.0251
10
102
103
Λ3HLL
mhHG
eVL
Λ2=0.02
Λ2=0.2
LEP
LEP Α=1
Α=1
0-0.005-0.01-0.015-0.02-0.025102
103
104
Λ3HLL
mH
2HG
eVL
Λ2=0.02
Λ2=0.2
LEPLEP
Α=1Α=1
0-0.005-0.01-0.015-0.02-0.025150
160
170
180
190
200
Λ3HLL
mH
1HG
eVL
Λ2=0.2
ΒΛ1=1
Λ1=0
LEP Α=1
0-0.005-0.01-0.015-0.02-0.025150
160
170
180
190
200
Λ3HLL
mH
1HG
eVL
Λ2=0.02
Β=0
ΒΛ1=1
Λ1=0
LEP Α=1
CERN, 16 June 2011 A. Pilaftsis
-
Extension by RH Neutrinos
LYν = − hνijL̄iLΦ̃ν0jR −1
2h
Nij ν̄
0CiR Sν
0jR −
1
2h̃
Nij ν̄
0iRSν
0CjR + H.c.
Neutrino mass spectrum:
LMassν = −1
2
(ν̄0iL, ν̄
0CiR
) ( 0 mDijm
TDij mMij
) (ν0CjLν0jR
)+ H.c.,
with
mD =φ√2
hν , mM =
1√2
[σ
(h
N + h̃N†)
+ iJ(h
N − h̃N†) ]
.
Type-II MSISM with σ ↔ J symmetry =⇒ hN = −ih̃N†.
mν = −1
4
√−λ3(Λ)λ1(Λ)
vSM hν(RehN)−1 hνT , mN = 2
√λ1(Λ)
−λ3(Λ)vSM Reh
N .
CERN, 16 June 2011 A. Pilaftsis
-
Extension by RH Neutrinos [L. Alexander–Nunneley, AP, JHEP09 (2010) 021]
0 0.05 0.1 0.15 0.20
200
400
600
800
Re hNHLL
mNHG
eVL
LEP
ΒΛ1=1
Λ1=0
Β=0
Case A
0 0.05 0.1 0.15 0.20
200
400
600
800
Re hNHLL
mNHG
eVL
LEPΒΛ1=1
Λ1=0
Β=0Α=1
Case B
0 0.05 0.1 0.15 0.20
200
400
600
800
Re hNHLL
mNHG
eVL
LEP
ΒΛ1=1
Λ1=0
Β=0
Case C
A: λ2(Λ) = 0.1 , λ3(Λ) = −0.01 ,
B: λ2(Λ) = 0.1 , λ3(Λ) = −0.005 ,
C: λ2(Λ) = 0.05 , λ3(Λ) = −0.005 .
CERN, 16 June 2011 A. Pilaftsis
-
• Conclusions
CERN, 16 June 2011 A. Pilaftsis
-
• Conclusions
• Perturbative SI extension(s) of the SM as potential solution tothe gauge hierarchy problem (up to quantum gravity effects)
CERN, 16 June 2011 A. Pilaftsis
-
• Conclusions
• Perturbative SI extension(s) of the SM as potential solution tothe gauge hierarchy problem (up to quantum gravity effects)
• MSISM = SI SM + 1 complex singlet scalar S
CERN, 16 June 2011 A. Pilaftsis
-
• Conclusions
• Perturbative SI extension(s) of the SM as potential solution tothe gauge hierarchy problem (up to quantum gravity effects)
• MSISM = SI SM + 1 complex singlet scalar S
• Convexity conditions and classification of MSISM flat directions
CERN, 16 June 2011 A. Pilaftsis
-
• Conclusions
• Perturbative SI extension(s) of the SM as potential solution tothe gauge hierarchy problem (up to quantum gravity effects)
• MSISM = SI SM + 1 complex singlet scalar S
• Convexity conditions and classification of MSISM flat directions
• 1-loop effective potential and radiative EW breaking of SI
CERN, 16 June 2011 A. Pilaftsis
-
• Conclusions
• Perturbative SI extension(s) of the SM as potential solution tothe gauge hierarchy problem (up to quantum gravity effects)
• MSISM = SI SM + 1 complex singlet scalar S
• Convexity conditions and classification of MSISM flat directions
• 1-loop effective potential and radiative EW breaking of SI
• EW constraints + perturbativity up to MPl=⇒ Type-II U(1)-violating MSISM + Z4 symmetry
=⇒ Stable H2 Scalar as DM + Maximal Spontaneous CP Violation
CERN, 16 June 2011 A. Pilaftsis
-
• Conclusions
• Perturbative SI extension(s) of the SM as potential solution tothe gauge hierarchy problem (up to quantum gravity effects)
• MSISM = SI SM + 1 complex singlet scalar S
• Convexity conditions and classification of MSISM flat directions
• 1-loop effective potential and radiative EW breaking of SI
• EW constraints + perturbativity up to MPl=⇒ Type-II U(1)-violating MSISM + Z4 symmetry
=⇒ Stable H2 Scalar as DM + Maximal Spontaneous CP Violation
• RH neutrinos predicted at the EW scale in the Type-II MSISM
CERN, 16 June 2011 A. Pilaftsis
-
• Conclusions
• Perturbative SI extension(s) of the SM as potential solution tothe gauge hierarchy problem (up to quantum gravity effects)
• MSISM = SI SM + 1 complex singlet scalar S
• Convexity conditions and classification of MSISM flat directions
• 1-loop effective potential and radiative EW breaking of SI
• EW constraints + perturbativity up to MPl=⇒ Type-II U(1)-violating MSISM + Z4 symmetry
=⇒ Stable H2 Scalar as DM + Maximal Spontaneous CP Violation
• RH neutrinos predicted at the EW scale in the Type-II MSISM
• LHC will partially probe the Higgs sector of the MSISM
CERN, 16 June 2011 A. Pilaftsis