heavy dark matter through the higgs portal€¦ · heavy dark matter through the higgs portal...
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Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Heavy Dark Matter Through The HiggsPortal
Stephen M. West
University of Oxford, Magdalen College
SHEP - 1st February 2008
J. March-Russell, SMW, D. Cumberbatch, D. Hooper:
arXiv:0801.3440 [hep-ph] and more to come...
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Outline
IntroductionHidden Sectors, Hidden Valleys and Higgs Portals.DM in Higgs Portal Models
Heavy DM from SUSY Higgs PortalsThe ModelThe Sommerfeld Enhancement
DM PhenomenolgyRelic Density CalculationDirect and Indirect Detection
Summary
Fat Higgs Appendix
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Outline
IntroductionHidden Sectors, Hidden Valleys and Higgs Portals.DM in Higgs Portal Models
Heavy DM from SUSY Higgs PortalsThe ModelThe Sommerfeld Enhancement
DM PhenomenolgyRelic Density CalculationDirect and Indirect Detection
Summary
Fat Higgs Appendix
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Outline
IntroductionHidden Sectors, Hidden Valleys and Higgs Portals.DM in Higgs Portal Models
Heavy DM from SUSY Higgs PortalsThe ModelThe Sommerfeld Enhancement
DM PhenomenolgyRelic Density CalculationDirect and Indirect Detection
Summary
Fat Higgs Appendix
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Outline
IntroductionHidden Sectors, Hidden Valleys and Higgs Portals.DM in Higgs Portal Models
Heavy DM from SUSY Higgs PortalsThe ModelThe Sommerfeld Enhancement
DM PhenomenolgyRelic Density CalculationDirect and Indirect Detection
Summary
Fat Higgs Appendix
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Outline
IntroductionHidden Sectors, Hidden Valleys and Higgs Portals.DM in Higgs Portal Models
Heavy DM from SUSY Higgs PortalsThe ModelThe Sommerfeld Enhancement
DM PhenomenolgyRelic Density CalculationDirect and Indirect Detection
Summary
Fat Higgs Appendix
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Hidden Sectors.
Hidden sectors common in many extensions of SM,e.g. SUSY, string theory, hidden/secluded RSwarped throats etc...
We consider, extra states not charged under the SM gauge group,
possibly charged under exotic gauge symmetry. SM or MSSM communicates with hidden sector via
Z ′ or Higgs interactions. Nice feature: interactions between hidden sector and
SM involve d ≤ 4 operators. Also consider mass scale of hidden sector close to or
just above weak scale. Attractive alternative to usual desert above weak
scale.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Hidden Sectors.
Hidden sectors common in many extensions of SM,e.g. SUSY, string theory, hidden/secluded RSwarped throats etc...
We consider, extra states not charged under the SM gauge group,
possibly charged under exotic gauge symmetry. SM or MSSM communicates with hidden sector via
Z ′ or Higgs interactions. Nice feature: interactions between hidden sector and
SM involve d ≤ 4 operators. Also consider mass scale of hidden sector close to or
just above weak scale. Attractive alternative to usual desert above weak
scale.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
SM-hidden sector communication viaZ ′ interactions.
Z ′ interactions occur either by: SM states charged under U(1)′ or Indirectly due to kinetic mixing term εF µν
Y F ′µν
(e.g. Babu, Kolda, March-Ruessll.)
Example Hidden Valley Model: SM gauge group, Gsm, extended by non-abelian
group Gv . SM states only charged under U(1)′ subgroup of Gv . Extra “valley" particles not charged under Gsm. Example Gv = U(1)′×SU(nv ). Interesting collider phenomenology and DM
possibilities.
(e.g. Strassler, Zurek; Han...)
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
SM-hidden sector communication viaZ ′ interactions.
Z ′ interactions occur either by: SM states charged under U(1)′ or Indirectly due to kinetic mixing term εF µν
Y F ′µν
(e.g. Babu, Kolda, March-Ruessll.)
Example Hidden Valley Model: SM gauge group, Gsm, extended by non-abelian
group Gv . SM states only charged under U(1)′ subgroup of Gv . Extra “valley" particles not charged under Gsm. Example Gv = U(1)′×SU(nv ). Interesting collider phenomenology and DM
possibilities.
(e.g. Strassler, Zurek; Han...)
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
SM-hidden sector communication via“Higgs Portals".
Extra states, not charged under Gsm but can becharged under exotic gauge symmetry.
Communication via interactions involving SM (orMSSM) Higgs
λ |H|2 |φ |2
(Schabinger, Wells; Patt, Wilczek...)
φ is a singlet under Gsm but charged under Ghidden. φ could have many more interactions with hidden
sector states. Changes Higgs phenomenology. Provides opportunities for DM in the hidden sector.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
SM-hidden sector communication via“Higgs Portals".
Extra states, not charged under Gsm but can becharged under exotic gauge symmetry.
Communication via interactions involving SM (orMSSM) Higgs
λ |H|2 |φ |2
(Schabinger, Wells; Patt, Wilczek...)
φ is a singlet under Gsm but charged under Ghidden. φ could have many more interactions with hidden
sector states. Changes Higgs phenomenology. Provides opportunities for DM in the hidden sector.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Can also have couplings involving fermions:
ψψLH
just neutrino Dirac term or in SUSY
W ⊃ λSHuHd
just the usual term found in models like the NMSSM. Point is ψ or S could have further interactions with
other states from the Hidden sector.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Can also have couplings involving fermions:
ψψLH
just neutrino Dirac term or in SUSY
W ⊃ λSHuHd
just the usual term found in models like the NMSSM. Point is ψ or S could have further interactions with
other states from the Hidden sector.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Can also have couplings involving fermions:
ψψLH
just neutrino Dirac term or in SUSY
W ⊃ λSHuHd
just the usual term found in models like the NMSSM. Point is ψ or S could have further interactions with
other states from the Hidden sector.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
DM in Higgs Portal Models.
Assume dominant form of DM comes from hiddensector,
Efficient coannihilation channels for neutralino LSP or Light gravitino LSP in gauge mediated models or Rp violation
Example 1: Direct coupling between hidden andvisible sector
λ |H|2 |φ |2 Coannihilation of hidden sector DM,
φ
φ
H
H
λ
(McDonald...)
(Cirelli, Fornengo, Strumia...)
DM with large masses (multi TeV) for large λinteresting parameter range.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
DM in Higgs Portal Models.
Assume dominant form of DM comes from hiddensector,
Efficient coannihilation channels for neutralino LSP or Light gravitino LSP in gauge mediated models or Rp violation
Example 1: Direct coupling between hidden andvisible sector
λ |H|2 |φ |2 Coannihilation of hidden sector DM,
φ
φ
H
H
λ
(McDonald...)
(Cirelli, Fornengo, Strumia...)
DM with large masses (multi TeV) for large λinteresting parameter range.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
DM in Higgs Portal Models.
Assume dominant form of DM comes from hiddensector,
Efficient coannihilation channels for neutralino LSP or Light gravitino LSP in gauge mediated models or Rp violation
Example 1: Direct coupling between hidden andvisible sector
λ |H|2 |φ |2 Coannihilation of hidden sector DM,
φ
φ
H
H
λ
(McDonald...)
(Cirelli, Fornengo, Strumia...)
DM with large masses (multi TeV) for large λinteresting parameter range.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Example 2: Indirect coupling between hidden andvisible sector
λ ′ψψφ + λφψvisψ ′vis
For SUSY modelψvisψ ′
vis ⇒ Higgsino pair, Higgsino Lepton pair. Coannihilation diagram:
ψ
ψ
ψ ′vis
ψvis
φλ ′ λ
ψ DM could be heavy requiring large couplings to beviable thermal relic.
For DM above ∼ 1TeV extra non-perturbativeSommerfeld effect important, more later...
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Example 2: Indirect coupling between hidden andvisible sector
λ ′ψψφ + λφψvisψ ′vis
For SUSY modelψvisψ ′
vis ⇒ Higgsino pair, Higgsino Lepton pair. Coannihilation diagram:
ψ
ψ
ψ ′vis
ψvis
φλ ′ λ
ψ DM could be heavy requiring large couplings to beviable thermal relic.
For DM above ∼ 1TeV extra non-perturbativeSommerfeld effect important, more later...
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Example 2: Indirect coupling between hidden andvisible sector
λ ′ψψφ + λφψvisψ ′vis
For SUSY modelψvisψ ′
vis ⇒ Higgsino pair, Higgsino Lepton pair. Coannihilation diagram:
ψ
ψ
ψ ′vis
ψvis
φλ ′ λ
ψ DM could be heavy requiring large couplings to beviable thermal relic.
For DM above ∼ 1TeV extra non-perturbativeSommerfeld effect important, more later...
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Heavy DM in SUSY Higgs Portal Models
The model defined by the superpotential
W=WMSSM (µ=0)+λNHuHd+t2N+ λ ′2 NS2+
ms2 S2
S has a non-R Z2 symmetry leads to a stable relic. DM particle will be s (fermionic cpt of S). Model is simple extension of MNSSM
(Panagiotakopoulos, Pilaftsis; Dedes et al.)
MNSSM coupling, λ , taken perturbative up to Mp. Our variant, added another singlet, S, with large
(multi TeV) mass. We need large λ and λ ′ to get correct relic
abundance.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Heavy DM in SUSY Higgs Portal Models
The model defined by the superpotential
W=WMSSM (µ=0)+λNHuHd+t2N+ λ ′2 NS2+
ms2 S2
S has a non-R Z2 symmetry leads to a stable relic. DM particle will be s (fermionic cpt of S). Model is simple extension of MNSSM
(Panagiotakopoulos, Pilaftsis; Dedes et al.)
MNSSM coupling, λ , taken perturbative up to Mp. Our variant, added another singlet, S, with large
(multi TeV) mass. We need large λ and λ ′ to get correct relic
abundance.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Heavy DM in SUSY Higgs Portal Models
The model defined by the superpotential
W=WMSSM (µ=0)+λNHuHd+t2N+ λ ′2 NS2+
ms2 S2
S has a non-R Z2 symmetry leads to a stable relic. DM particle will be s (fermionic cpt of S). Model is simple extension of MNSSM
(Panagiotakopoulos, Pilaftsis; Dedes et al.)
MNSSM coupling, λ , taken perturbative up to Mp. Our variant, added another singlet, S, with large
(multi TeV) mass. We need large λ and λ ′ to get correct relic
abundance.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Heavy DM in SUSY Higgs Portal Models
The model defined by the superpotential
W=WMSSM (µ=0)+λNHuHd+t2N+ λ ′2 NS2+
ms2 S2
S has a non-R Z2 symmetry leads to a stable relic. DM particle will be s (fermionic cpt of S). Model is simple extension of MNSSM
(Panagiotakopoulos, Pilaftsis; Dedes et al.)
MNSSM coupling, λ , taken perturbative up to Mp. Our variant, added another singlet, S, with large
(multi TeV) mass. We need large λ and λ ′ to get correct relic
abundance.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Large coupling and UV completion
Taking cut off for theory Λ0 ≥ MGUT⇒ Couplings, λ ,λ ′, blow up well before Λ0.⇒ S mass term implies large tadpole terms spoilingthe stability of the weak scale. (Panagiotakopoulos, Pilaftsis; Dedes...)
Can take S to be elementary and simultaneouslychoose large values for λ ,λ ′ if theory is an effectivetheory with cutoff Λ0 < 100TeV.
No problem in having a mass for S in this case.But no UV completion
Treat model as effective theory of the “Fat Higgs"model (Harnik et al.)
S ⇒ composite meson field of new susy-preservingstrong-interaction dynamics.
No large tadpole terms. Natural to expect large couplings λ and λ ′.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Large coupling and UV completion
Taking cut off for theory Λ0 ≥ MGUT⇒ Couplings, λ ,λ ′, blow up well before Λ0.⇒ S mass term implies large tadpole terms spoilingthe stability of the weak scale. (Panagiotakopoulos, Pilaftsis; Dedes...)
Can take S to be elementary and simultaneouslychoose large values for λ ,λ ′ if theory is an effectivetheory with cutoff Λ0 < 100TeV.
No problem in having a mass for S in this case.But no UV completion
Treat model as effective theory of the “Fat Higgs"model (Harnik et al.)
S ⇒ composite meson field of new susy-preservingstrong-interaction dynamics.
No large tadpole terms. Natural to expect large couplings λ and λ ′.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Large coupling and UV completion
Taking cut off for theory Λ0 ≥ MGUT⇒ Couplings, λ ,λ ′, blow up well before Λ0.⇒ S mass term implies large tadpole terms spoilingthe stability of the weak scale. (Panagiotakopoulos, Pilaftsis; Dedes...)
Can take S to be elementary and simultaneouslychoose large values for λ ,λ ′ if theory is an effectivetheory with cutoff Λ0 < 100TeV.
No problem in having a mass for S in this case.But no UV completion
Treat model as effective theory of the “Fat Higgs"model (Harnik et al.)
S ⇒ composite meson field of new susy-preservingstrong-interaction dynamics.
No large tadpole terms. Natural to expect large couplings λ and λ ′.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
DM Interactions Lagrangian determining relevant interactions and
masses, L⊃Lfermion+Lscalar, where
Lfermion =(
−λnhu hd−λ nhuhd−λ nhu hd− λ ′2 nss−λ ′nss−ms
2 ss+h.c)
Lscalar= |λ ′ns+mss|2+|λhuhd+t2+(λ ′/2)s2|2+|λnhd |2+|λnhu |2
+(
Aλ λnhuhd+Aλ ′
2 λ ′ns2+ B2 mss2+Ct2n+h.c
)
+M2hu|hu |2+M2
hd|hd |2
+M2s |s|2+M2
n |n|2,
In the limit ms ≫ msusy, mass differencems −ms ≃ m2
susy/ms < Tf ≃ ms/25.⇒ s and s will freeze out at similar T⇒ must include s annihilations in relic densitycalculation
Relevant scalar interactions:λ ′ms |s|2 n +h.c,
λ ′λhuhds∗2 +h.c
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
DM Interactions Lagrangian determining relevant interactions and
masses, L⊃Lfermion+Lscalar, where
Lfermion =(
−λnhu hd−λ nhuhd−λ nhu hd− λ ′2 nss−λ ′nss−ms
2 ss+h.c)
Lscalar= |λ ′ns+mss|2+|λhuhd+t2+(λ ′/2)s2|2+|λnhd |2+|λnhu |2
+(
Aλ λnhuhd+Aλ ′
2 λ ′ns2+ B2 mss2+Ct2n+h.c
)
+M2hu|hu |2+M2
hd|hd |2
+M2s |s|2+M2
n |n|2,
In the limit ms ≫ msusy, mass differencems −ms ≃ m2
susy/ms < Tf ≃ ms/25.⇒ s and s will freeze out at similar T⇒ must include s annihilations in relic densitycalculation
Relevant scalar interactions:λ ′ms |s|2 n +h.c,
λ ′λhuhds∗2 +h.c
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
The Sommerfeld Enhancement
Heavy DM moving at small (relative) velocities, dueto exchange of scalar states⇒ Enhancement ∝ 1/v .
(Sommerfeld, Hisano et al; Strumia et al...)
Sommerfeld enhancement corresponds to thesummation of diagrams
Only significant for s-wave,(AM barrier suppresses effect).
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
The Sommerfeld Enhancement
Heavy DM moving at small (relative) velocities, dueto exchange of scalar states⇒ Enhancement ∝ 1/v .
(Sommerfeld, Hisano et al; Strumia et al...)
Sommerfeld enhancement corresponds to thesummation of diagrams
Only significant for s-wave,(AM barrier suppresses effect).
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Sommerfeld Enhancement
Calculation formulated as non-relativistic two-bodyQM problem with a potential.
Equivalent to the distorted Born-wave approximationcommon in nuclear physics.
Can be an enhancement or suppression for vectorstates.
Including the effect (to a good approximation)
σ = Rσ ℓ=0tree
Full calculation of R can be involved. For Yukawa potential, cannot be solved analytically.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Calculation of the Sommerfeld Enhancement
Our model, only scalar n states are “rungs on theladder"
The Schrödinger equation for the two dark matterparticle state, ψ ,
− 1ms
d2ψdr2 +V .ψ = K ψ , V = − λ ′2
8πr e−mnr , K = msv2
Enhancement factor is
R = |ψ(0)/ψ(∞)|2 .
Outgoing b.c., ψ ′(∞)/ψ(∞) = imsv
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Calculation of the Sommerfeld Enhancement
Our model, only scalar n states are “rungs on theladder"
The Schrödinger equation for the two dark matterparticle state, ψ ,
− 1ms
d2ψdr2 +V .ψ = K ψ , V = − λ ′2
8πr e−mnr , K = msv2
Enhancement factor is
R = |ψ(0)/ψ(∞)|2 .
Outgoing b.c., ψ ′(∞)/ψ(∞) = imsv
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Calculation of the Sommerfeld Enhancement
Our model, only scalar n states are “rungs on theladder"
The Schrödinger equation for the two dark matterparticle state, ψ ,
− 1ms
d2ψdr2 +V .ψ = K ψ , V = − λ ′2
8πr e−mnr , K = msv2
Enhancement factor is
R = |ψ(0)/ψ(∞)|2 .
Outgoing b.c., ψ ′(∞)/ψ(∞) = imsv
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Coulomb limit of Sommerfeld Enhancement
Recall approximation ms ≫ msusy
Analytic form for R in limit ε ≡ mn/ms = 0,
R =y
1−e−y , y =λ ′2
8v=
λ ′2
4vr
Small vr limit
R ≈ λ ′2
4vr
Next paper, more detailed analysis for Sommerfeldeffect, e.g. non-zero ε .
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Coulomb limit of Sommerfeld Enhancement
Recall approximation ms ≫ msusy
Analytic form for R in limit ε ≡ mn/ms = 0,
R =y
1−e−y , y =λ ′2
8v=
λ ′2
4vr
Small vr limit
R ≈ λ ′2
4vr
Next paper, more detailed analysis for Sommerfeldeffect, e.g. non-zero ε .
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
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DMPhenomenolgyRelic Density Calculation
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Summary
Fat HiggsAppendix
Relic Density Calculation
Important points and assumptions:
Take DM masses greater than ∼ 3TeV⇒ Tf ∼ ms
25 > Tc ⇒ EW still good symmetry,⇒ Apart from DM all fermions massless.
Set soft params to zero, Aλ = Aλ ′ . Soft masses negligible compared to ms.
Recall scalar s (ms ∼ ms +m2susy/ms)
⇒ s freezes out at similar T as s⇒ must include s annihilation rates in relic calc.
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IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
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DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Relic Density Calculation
Important points and assumptions:
Take DM masses greater than ∼ 3TeV⇒ Tf ∼ ms
25 > Tc ⇒ EW still good symmetry,⇒ Apart from DM all fermions massless.
Set soft params to zero, Aλ = Aλ ′ . Soft masses negligible compared to ms.
Recall scalar s (ms ∼ ms +m2susy/ms)
⇒ s freezes out at similar T as s⇒ must include s annihilation rates in relic calc.
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IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
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Summary
Fat HiggsAppendix
Scalar s annihilations. Interactions: λnhuhd + λ ′ms |s|2 n + λ ′λhuhds∗2+h.c.
Scalar states split into CP eigenstates, A= 1√2(φ+ia)
s∗
s
hu
hd
φn2λ′ms λ
s
s
h∗u
h∗d
λλ′
2
s∗
s∗
hu
hd
λλ′
2
s∗
s∗
φn
σ(ss → XX′) =
(λ′λ)2
256πvrm2s
y
1 − e−y, y =
λ′2
4vr
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ss annihilations. Interactions: λ nhuhd + λ nhuhd + λ ′nss+h.c
s
s
hu,hd
hd,hu
nλ′ λ
s
s
φnn
σ(ss → XX′) =
(λ′λ)2
2πvrm2s
y
1 − e−y, y =
λ′2
4vr
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ss annihilations. Interactions: λ ′
2 nss + λnhuhd+h.c
s
s
hu
hd
aniλ′γ5 iλγ5
s
s
φnan
σ(ss → XX′) =
(λ′λ)2
128πvrm2s
y
1 − e−y, y =
λ′2
4vr
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IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
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Summary
Fat HiggsAppendix
Relic Density Numerics.
Re-label (s,s) as (s1,s2) Define
ri≡gi (1+∆i )
3/2 exp[−x∆i ]geff
, ∆i=(mi−m1)/m1,
where
geff=∑i gi (1+∆i)3/2 exp[−x∆i ], x=ms/T
For our two species
g1=g2=2, ∆1=0, ∆2=ms−ms≃m2
susyms
.
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IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
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Summary
Fat HiggsAppendix
Relic Density Numerics.
Re-label (s,s) as (s1,s2) Define
ri≡gi (1+∆i )
3/2 exp[−x∆i ]geff
, ∆i=(mi−m1)/m1,
where
geff=∑i gi (1+∆i)3/2 exp[−x∆i ], x=ms/T
For our two species
g1=g2=2, ∆1=0, ∆2=ms−ms≃m2
susyms
.
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IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
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DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Relic Density Numerics.
Re-label (s,s) as (s1,s2) Define
ri≡gi (1+∆i )
3/2 exp[−x∆i ]geff
, ∆i=(mi−m1)/m1,
where
geff=∑i gi (1+∆i)3/2 exp[−x∆i ], x=ms/T
For our two species
g1=g2=2, ∆1=0, ∆2=ms−ms≃m2
susyms
.
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Summary
Fat HiggsAppendix
Freeze-out temperature, Tf = ms/xf , found byiteratively solving
xf =ln
[
0.038geffMpl ms〈σij vr〉√g⋆xf
]
,
with
σeff=∑i,j σij ri rj=∑2i,j σij
gi gjg2
eff(1+∆i )
3/2(1+∆j)3/2 exp(−x(∆i +∆j)).
Final relic density
Ωh2=1.07×109xf√g⋆Mpl (GeV)J , J=
∫ ∞xf
x−2aeffdx
(Griest, Seckel; Gondolo, Gelmini)
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Summary
Fat HiggsAppendix
Comparing with and without Sommerfeld.
Plotted for λ = λ ′ = 1.5,2,2.5, with msusy= 100GeV
5 10 15 20 25 30m
s~
0.1
0.2
0.3
0.4
0.5
0.6
Wh2
Without Sommerfeld
With Sommerfeld
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Fat HiggsAppendix
λ λ ′ parameter space. Plotted for msusy= 100GeV.
0.5 1.0 1.5 2.0 2.5 3.0Λ'
0.5
1.0
1.5
2.0
2.5
3.0
Λ
ms~= 23 TeV
ms~= 19 TeV
ms~= 15 TeV
ms~= 11 TeV
ms~= 7 TeV
ms~= 3 TeV
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Summary
Fat HiggsAppendix
Direct Detection. Main analyses next paper...
Quick look: Direct detection phenomenology: No restrictive limits
BUT sizeable fraction of future parameter spacecovered by next generation detectors.
Our DM interacts with quarks in nuclei via effectivescalar interaction
L = ∑U=u,c,t
CU ssUU + ∑D=d ,s,b
CDssDD,
where
CU = ∑i
λUV1iV2iλ ′
2m2hi
, CD = ∑i
λDV1iV3iλ ′
2m2hi
,
Vij is Higgs mixing matrix. (Griest; Bertone, Hooper, Silk...)
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IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
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Summary
Fat HiggsAppendix
Direct Detection. Main analyses next paper...
Quick look: Direct detection phenomenology: No restrictive limits
BUT sizeable fraction of future parameter spacecovered by next generation detectors.
Our DM interacts with quarks in nuclei via effectivescalar interaction
L = ∑U=u,c,t
CU ssUU + ∑D=d ,s,b
CDssDD,
where
CU = ∑i
λUV1iV2iλ ′
2m2hi
, CD = ∑i
λDV1iV3iλ ′
2m2hi
,
Vij is Higgs mixing matrix. (Griest; Bertone, Hooper, Silk...)
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IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
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Summary
Fat HiggsAppendix
Direct Detection Cont...
Estimate elastic scattering cross secton per nucleon
σsN ∼ 2×10−7 pb(
Vij
0.5
)4 (
λ ′
3
)2 (
120GeVmh1
)4
.
For ms ≥ 3TeV, cross section below currentconstraints from experiments such as XENON andCDMS.
Future measurements could probe parameter region. For smaller values of λ ′,Vij gets harder...
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Summary
Fat HiggsAppendix
Indirect Detection.
Indirect signals are modified by potentially largeSommerfeld enhancements.
Depends on astrophysical environment, specificallyvelocity distribution.
Annihilation rates can be enhanced by 103 to 105 ormore.
Could favour objects with low velocity dispersion e.g.dwarf satellite galaxies of the Milky Way.
Full calculation needed: Need to understand resonance structure of the
Sommerfeld enhancement in the non-coulombic andlow vr regime
Need full details of low energy model.
Potential for significant signals in future observations.
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IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
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Heavy DM fromSUSY HiggsPortalsThe Model
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Summary
Fat HiggsAppendix
Indirect Detection.
Indirect signals are modified by potentially largeSommerfeld enhancements.
Depends on astrophysical environment, specificallyvelocity distribution.
Annihilation rates can be enhanced by 103 to 105 ormore.
Could favour objects with low velocity dispersion e.g.dwarf satellite galaxies of the Milky Way.
Full calculation needed: Need to understand resonance structure of the
Sommerfeld enhancement in the non-coulombic andlow vr regime
Need full details of low energy model.
Potential for significant signals in future observations.
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IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Indirect Detection.
Indirect signals are modified by potentially largeSommerfeld enhancements.
Depends on astrophysical environment, specificallyvelocity distribution.
Annihilation rates can be enhanced by 103 to 105 ormore.
Could favour objects with low velocity dispersion e.g.dwarf satellite galaxies of the Milky Way.
Full calculation needed: Need to understand resonance structure of the
Sommerfeld enhancement in the non-coulombic andlow vr regime
Need full details of low energy model.
Potential for significant signals in future observations.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Summary Models with very heavy (∼ 3−30TeV ) dark matter. Motivated by Higgs Portal (and Hidden Valley)
models⇒ DM interacts with visible sector only throughHiggs interactions (and Z ′).
Such models have UV completion using a composite“Fat Higgs” model.
DM annihilations are enhanced by non-perturbativeSommerfeld effect.
DM up to ∼30 TeV can be thermal relic with correctrelic abundance.
Our DM potentially detectable by direct and indirectDM experiments.
Sommerfeld corrections can dramatically enhanceDM annihilation rate for low velocities.⇒ Greatly improving detection possibilities.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Summary Models with very heavy (∼ 3−30TeV ) dark matter. Motivated by Higgs Portal (and Hidden Valley)
models⇒ DM interacts with visible sector only throughHiggs interactions (and Z ′).
Such models have UV completion using a composite“Fat Higgs” model.
DM annihilations are enhanced by non-perturbativeSommerfeld effect.
DM up to ∼30 TeV can be thermal relic with correctrelic abundance.
Our DM potentially detectable by direct and indirectDM experiments.
Sommerfeld corrections can dramatically enhanceDM annihilation rate for low velocities.⇒ Greatly improving detection possibilities.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Summary Models with very heavy (∼ 3−30TeV ) dark matter. Motivated by Higgs Portal (and Hidden Valley)
models⇒ DM interacts with visible sector only throughHiggs interactions (and Z ′).
Such models have UV completion using a composite“Fat Higgs” model.
DM annihilations are enhanced by non-perturbativeSommerfeld effect.
DM up to ∼30 TeV can be thermal relic with correctrelic abundance.
Our DM potentially detectable by direct and indirectDM experiments.
Sommerfeld corrections can dramatically enhanceDM annihilation rate for low velocities.⇒ Greatly improving detection possibilities.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
Summary Models with very heavy (∼ 3−30TeV ) dark matter. Motivated by Higgs Portal (and Hidden Valley)
models⇒ DM interacts with visible sector only throughHiggs interactions (and Z ′).
Such models have UV completion using a composite“Fat Higgs” model.
DM annihilations are enhanced by non-perturbativeSommerfeld effect.
DM up to ∼30 TeV can be thermal relic with correctrelic abundance.
Our DM potentially detectable by direct and indirectDM experiments.
Sommerfeld corrections can dramatically enhanceDM annihilation rate for low velocities.⇒ Greatly improving detection possibilities.
Heavy Dark MatterThrough TheHiggs Portal
IntroductionHidden Sectors, HiddenValleys and Higgs Portals.
DM in Higgs Portal Models
Heavy DM fromSUSY HiggsPortalsThe Model
The SommerfeldEnhancement
DMPhenomenolgyRelic Density Calculation
Direct and IndirectDetection
Summary
Fat HiggsAppendix
The Fat Higgs Model.
The Fat Higgs model is an N = 1 supersymmetricSU(2) gauge theory with six doublets.
Full symmetry is SU(2)L ×SU(2)H gauge andSU(2)R ×SU(2)g ×U(1)R global,U(1)Y subgroup of SU(2)R is also gauged.
SU(2)H becomes strongly coupled at some scale, ΛH
Six doublets charged under SU(2)H form compositeobjects, Mij = TiTj , with (i, j=1...6)
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Fat HiggsAppendix
The Fat Higgs Model.
The Fat Higgs model is an N = 1 supersymmetricSU(2) gauge theory with six doublets.
Full symmetry is SU(2)L ×SU(2)H gauge andSU(2)R ×SU(2)g ×U(1)R global,U(1)Y subgroup of SU(2)R is also gauged.
SU(2)H becomes strongly coupled at some scale, ΛH
Six doublets charged under SU(2)H form compositeobjects, Mij = TiTj , with (i, j=1...6)
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Fat HiggsAppendix
Particle content of the Fat Higgs Model:
Field SU(2)L SU(2)H SU(2)R SU(2)g U(1)R Z2
T1 2 2 1 1 0 +T2 2 2 1 1 0 -T3 1 2 2 1 1 -T4 1 2 2 1 1 +T5 1 2 1 2 1 +T6 1 2 1 2 1 +P11 2 1 1 2 1 +P12 2 1 1 2 1 +P21 2 1 1 2 1 -P22 2 1 1 2 1 -Q11 1 1 2 2 1 -Q12 1 1 2 2 1 -Q21 1 1 2 2 1 +Q22 1 1 2 2 1 +Sa 1 1 1 1 2 -Sb 1 1 1 1 2 -
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More Fat Higgs... Tree Level Superpotential, WFHtot = W1 +W2, where
W1 = y1SaT1T2+y2SbT3T4+y3SaT3T4+y4SbT1T2W2−mT5T6
W2 = y5
(
T1 T2
)
P
(
T5
T6
)
+ y6
(
T3 T4
)
Q
(
T5
T6
)
.
After SU(2)H becomes strong, effectivesuperpotential becomes (after canon. norm.)
Wdyn = λ(PfM−v20 M56)+m1SaM12+m2SbM34+m3SaM34+m4SbM12
+ m5(M15P11+M16P12+M25P21+M26P22)
+ m6(M35Q11+M36Q12+M45Q21+M36Q22),
Using NDA , we have (Luty)
v20 ∼ mΛH
(4π)2,
mi ∼ yiΛH4π ,
λ(ΛH ) ∼ 4π.
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Fat HiggsAppendix
More Fat Higgs... Tree Level Superpotential, WFHtot = W1 +W2, where
W1 = y1SaT1T2+y2SbT3T4+y3SaT3T4+y4SbT1T2W2−mT5T6
W2 = y5
(
T1 T2
)
P
(
T5
T6
)
+ y6
(
T3 T4
)
Q
(
T5
T6
)
.
After SU(2)H becomes strong, effectivesuperpotential becomes (after canon. norm.)
Wdyn = λ(PfM−v20 M56)+m1SaM12+m2SbM34+m3SaM34+m4SbM12
+ m5(M15P11+M16P12+M25P21+M26P22)
+ m6(M35Q11+M36Q12+M45Q21+M36Q22),
Using NDA , we have (Luty)
v20 ∼ mΛH
(4π)2,
mi ∼ yiΛH4π ,
λ(ΛH ) ∼ 4π.
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Summary
Fat HiggsAppendix
Yet More Fat Higgs... Make assumption (m5,m6) ≫ (m1,m2,m3,m4),
integrate out heavy states
W ′dyn = λM56(M14M23−M24M13−v2
0 +M12M34)
+ m1SaM12+m2SbM34+m3SaM34+m4SbM12.
With m1 ∼ m2 ∼ m3 ∼ m4 ∼ m′, fermioniccomponents of Sa,Sb,M12 and M34 mix lightesteigenvalue ∼ m′.
Diagonalization, integrate out all but the lightesteigenvalue, S1,
W=λN(HuHd−v20)+
λ ′2 NS2
1+ms1
2 S21,
where(
H+u
H0u
)
=
(
M13
M23
)
,
(
H0d
H−d
)
=
(
M14
M24
)
,N=M56.
Final assumption: ms1 ≫electroweak scale and softsupersymmetry breaking masses.
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Summary
Fat HiggsAppendix
Yet More Fat Higgs... Make assumption (m5,m6) ≫ (m1,m2,m3,m4),
integrate out heavy states
W ′dyn = λM56(M14M23−M24M13−v2
0 +M12M34)
+ m1SaM12+m2SbM34+m3SaM34+m4SbM12.
With m1 ∼ m2 ∼ m3 ∼ m4 ∼ m′, fermioniccomponents of Sa,Sb,M12 and M34 mix lightesteigenvalue ∼ m′.
Diagonalization, integrate out all but the lightesteigenvalue, S1,
W=λN(HuHd−v20)+
λ ′2 NS2
1+ms1
2 S21,
where(
H+u
H0u
)
=
(
M13
M23
)
,
(
H0d
H−d
)
=
(
M14
M24
)
,N=M56.
Final assumption: ms1 ≫electroweak scale and softsupersymmetry breaking masses.