This document presents an overview of Alexander Anderson-Natale's work on radiative neutrino mass generation, dark matter, and flavor symmetries. It discusses the scotogenic model, where neutrino mass arises from dark matter loops. It also covers neutrino oscillation parameters described by the PMNS matrix, flavor symmetries like A4 and Δ(27) that can explain mixing patterns, and the possibility of probing scotogenic models at colliders through signatures of warm dark matter or color-triplet scalars. The document is divided into sections on neutrino mass and symmetries versus collider signatures of these models.
1. Radiative Neutrino Mass, Dark Matter, Flavor
Symmetry, and Collider Signatures
Alexander Anderson-Natale
University of California Riverside
July 7th, 2015
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2. Acknowledgements
Dr. Ernest Ma, Dr. Jose Wudka, Dr. Hai-Bo Yu
Dr. Subhaditya Bhattacharya, Dr. Daniel Wegman, Dr. Ahmed Rashed,
Dr. Kiel Howe, Sean Fraser, Corey Kownacki, Nick Pollard, Oleg Popov,
Mohammadreza Zakeri
Kayleigh Anderson-Natale,
Nick Natale, Jennifer Swanberg, Jodey Anderson, Michael Anderson,
Cooper Jaquish, Jessica Jaquish, Sara Anderson, Andy Schou, Hope
Humphries, Peg Swanberg
A special thanks to all my friends (particularly ’gym squad’), colleagues,
and mentors over the years and the rest of the faculty and staff at
UCRiverside.
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3. Topics cover 5 of my 6 published papers.
Divided into two categories:
(I) Neutrino oscillation & symmetries
(II) Scotogenic collider signatures
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12. Introduction
e
νe
d
u
µ
νµ
s
c
τ
ντ
b
t
h
g
W±
Z
γLeptonsQuarks
have mass, but too light
Only clear evidence of new particle physics.
Understanding these phenomena a continued interest for physicists.
M. Schumann, Braz. J. Phys. 44 (2014) 483-493, arXiv:1310.5217 [astro-ph.CO].
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13. The Scotogenic Model
νi νjNk
η0
η0
φ0
φ0
×
Minimal scotogenic model.
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14. The Scotogenic Model
From Greek ’scoto’ for darkness: mν generated via DM interactions
νi νjNk
η0
η0
φ0
φ0
×
Minimal scotogenic model.
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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15. The Scotogenic Model
From Greek ’scoto’ for darkness: mν generated via DM interactions
Additional Z2 symmetry (’dark’ symmetry)
νi νjNk
η0
η0
φ0
φ0
×
Minimal scotogenic model.
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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16. The Scotogenic Model
From Greek ’scoto’ for darkness: mν generated via DM interactions
Additional Z2 symmetry (’dark’ symmetry)
Inert (dark) scalar doublet η
νi νjNk
η0
η0
φ0
φ0
×
Minimal scotogenic model.
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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17. The Scotogenic Model
From Greek ’scoto’ for darkness: mν generated via DM interactions
Additional Z2 symmetry (’dark’ symmetry)
Inert (dark) scalar doublet η
Three neutral fermion singlets Ni
νi νjNk
η0
η0
φ0
φ0
×
Minimal scotogenic model.
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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18. The Scotogenic Model
From Greek ’scoto’ for darkness: mν generated via DM interactions
Additional Z2 symmetry (’dark’ symmetry)
Inert (dark) scalar doublet η
Three neutral fermion singlets Ni
νi νjNk
η0
η0
φ0
φ0
×
Minimal scotogenic model.
Mν =
k
hikhjkmNk
16π2
m2
R
m2
R − m2
Nk
log(m2
R/m2
Nk
) −
m2
I
m2
I − m2
Nk
log(m2
I/m2
Nk
)
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19. Neutrino Oscillation & Mass
The evidence for the existence of neutrino mass is thoroughly established,
and is described by 3 angles and (potentially) 3 phases in form of
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix (Uαi):
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20. Neutrino Oscillation & Mass
The evidence for the existence of neutrino mass is thoroughly established,
and is described by 3 angles and (potentially) 3 phases in form of
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix (Uαi):
c12c13 s12c13 s13e−iδcp
−s12c23 − c12s23s13eiδcp c12c23 − s12s23s13eiδcp s23c13
s12s23 − c12c23s13eiδcp −c12s23 − s12c23s13eiδcp c23c13
× diag(1, eiα21/2
, eiα31/2
),
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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21. Neutrino Oscillation & Mass
The evidence for the existence of neutrino mass is thoroughly established,
and is described by 3 angles and (potentially) 3 phases in form of
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix (Uαi):
c12c13 s12c13 s13e−iδcp
−s12c23 − c12s23s13eiδcp c12c23 − s12s23s13eiδcp s23c13
s12s23 − c12c23s13eiδcp −c12s23 − s12c23s13eiδcp c23c13
× diag(1, eiα21/2
, eiα31/2
),
With 1 Dirac, 2 relative Majorana phases.
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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22. Neutrino Oscillation & Mass
The evidence for the existence of neutrino mass is thoroughly established,
and is described by 3 angles and (potentially) 3 phases in form of
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix (Uαi):
c12c13 s12c13 s13e−iδcp
−s12c23 − c12s23s13eiδcp c12c23 − s12s23s13eiδcp s23c13
s12s23 − c12c23s13eiδcp −c12s23 − s12c23s13eiδcp c23c13
× diag(1, eiα21/2
, eiα31/2
),
With 1 Dirac, 2 relative Majorana phases.
ν could be own anti-particle (Majorana), or Dirac fermion, if Majorana can
lead to ν-less double-beta decay with ’rate’:
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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23. Neutrino Oscillation & Mass
The evidence for the existence of neutrino mass is thoroughly established,
and is described by 3 angles and (potentially) 3 phases in form of
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix (Uαi):
c12c13 s12c13 s13e−iδcp
−s12c23 − c12s23s13eiδcp c12c23 − s12s23s13eiδcp s23c13
s12s23 − c12c23s13eiδcp −c12s23 − s12c23s13eiδcp c23c13
× diag(1, eiα21/2
, eiα31/2
),
With 1 Dirac, 2 relative Majorana phases.
ν could be own anti-particle (Majorana), or Dirac fermion, if Majorana can
lead to ν-less double-beta decay with ’rate’:
mee = |m1U2
e1 + m2U2
e2 + m3U2
e3|
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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24. Neutrino Oscillation & Mass
The evidence for the existence of neutrino mass is thoroughly established,
and is described by 3 angles and (potentially) 3 phases in form of
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix (Uαi):
c12c13 s12c13 s13e−iδcp
−s12c23 − c12s23s13eiδcp c12c23 − s12s23s13eiδcp s23c13
s12s23 − c12c23s13eiδcp −c12s23 − s12c23s13eiδcp c23c13
× diag(1, eiα21/2
, eiα31/2
),
With 1 Dirac, 2 relative Majorana phases.
ν could be own anti-particle (Majorana), or Dirac fermion, if Majorana can
lead to ν-less double-beta decay with ’rate’:
mee = |m1U2
e1 + m2U2
e2 + m3U2
e3|
CP phases & nature of ν unknown
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25. Flavor Symmetry
Observed ν oscillation close to Tribimaximal:
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26. Flavor Symmetry
Observed ν oscillation close to Tribimaximal:
2/3 1/3 0
1/6 1/3 1/2
1/6 1/3 1/2
, (1)
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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27. Flavor Symmetry
Observed ν oscillation close to Tribimaximal:
2/3 1/3 0
1/6 1/3 1/2
1/6 1/3 1/2
, (1)
Naturally accomplished in models with A4 flavor symmetry, θ13 = 0 can
be accommodated and is even predicted for certain models w/ ∆(27).
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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28. Flavor Symmetry
Observed ν oscillation close to Tribimaximal:
2/3 1/3 0
1/6 1/3 1/2
1/6 1/3 1/2
, (1)
Naturally accomplished in models with A4 flavor symmetry, θ13 = 0 can
be accommodated and is even predicted for certain models w/ ∆(27).
Basic methodology: miss-match between l and ν achieved via different
reps. of discrete group for lR and (ν, l)L
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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29. Flavor Symmetry
Observed ν oscillation close to Tribimaximal:
2/3 1/3 0
1/6 1/3 1/2
1/6 1/3 1/2
, (1)
Naturally accomplished in models with A4 flavor symmetry, θ13 = 0 can
be accommodated and is even predicted for certain models w/ ∆(27).
Basic methodology: miss-match between l and ν achieved via different
reps. of discrete group for lR and (ν, l)L → accommodates large
difference in ml while also providing the symmetry in ν oscillation (2 large
angles, 1 ’small’ angle).
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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30. Flavor Symmetry
Observed ν oscillation close to Tribimaximal:
2/3 1/3 0
1/6 1/3 1/2
1/6 1/3 1/2
, (1)
Naturally accomplished in models with A4 flavor symmetry, θ13 = 0 can
be accommodated and is even predicted for certain models w/ ∆(27).
Basic methodology: miss-match between l and ν achieved via different
reps. of discrete group for lR and (ν, l)L → accommodates large
difference in ml while also providing the symmetry in ν oscillation (2 large
angles, 1 ’small’ angle).
θ13 = 0 thus TBM not exactly correct, so rotate to this basis and
remaining terms are small corrections to TBM.
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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31. Extended Scotogenic Models
The basic scotogenic model has been studied in many contexts:
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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32. Extended Scotogenic Models
The basic scotogenic model has been studied in many contexts:
Addition of flavor symmetries
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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33. Extended Scotogenic Models
The basic scotogenic model has been studied in many contexts:
Addition of flavor symmetries ← avoids addition of many Higgses
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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34. Extended Scotogenic Models
The basic scotogenic model has been studied in many contexts:
Addition of flavor symmetries ← avoids addition of many Higgses
Expanded particle content for U(1)D DM (SIMPs)
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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35. Extended Scotogenic Models
The basic scotogenic model has been studied in many contexts:
Addition of flavor symmetries ← avoids addition of many Higgses
Expanded particle content for U(1)D DM (SIMPs)
Alternative particle content (doublet fermions, singlet scalars, etc.)
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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36. Extended Scotogenic Models
The basic scotogenic model has been studied in many contexts:
Addition of flavor symmetries ← avoids addition of many Higgses
Expanded particle content for U(1)D DM (SIMPs)
Alternative particle content (doublet fermions, singlet scalars, etc.)
SU(5) unification
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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37. Extended Scotogenic Models
The basic scotogenic model has been studied in many contexts:
Addition of flavor symmetries ← avoids addition of many Higgses
Expanded particle content for U(1)D DM (SIMPs)
Alternative particle content (doublet fermions, singlet scalars, etc.)
SU(5) unification
Unified scotogenic framework for quark & lepton masses
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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38. Extended Scotogenic Models
The basic scotogenic model has been studied in many contexts:
Addition of flavor symmetries ← avoids addition of many Higgses
Expanded particle content for U(1)D DM (SIMPs)
Alternative particle content (doublet fermions, singlet scalars, etc.)
SU(5) unification
Unified scotogenic framework for quark & lepton masses
Two-loop scotogenic mechanisms
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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39. Extended Scotogenic Models
The basic scotogenic model has been studied in many contexts:
Addition of flavor symmetries ← avoids addition of many Higgses
Expanded particle content for U(1)D DM (SIMPs)
Alternative particle content (doublet fermions, singlet scalars, etc.)
SU(5) unification
Unified scotogenic framework for quark & lepton masses
Two-loop scotogenic mechanisms
Other extended gauge groups (U(1)B−L, SU(6), SU(2)R × SU(2)L,
etc.)
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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40. Radiative Seesaw with A4
Minimal scotogenic model with ∆m2
η m2
ηR
, m2
ηI
m2
Nk
:
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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41. Radiative Seesaw with A4
Minimal scotogenic model with ∆m2
η m2
ηR
, m2
ηI
m2
Nk
:
→ (Mν)ij =
λ5v2
8π2
k
hijhjk
mNk
log
m2
Nk
m2
0
− 1 ,
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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42. Radiative Seesaw with A4
Minimal scotogenic model with ∆m2
η m2
ηR
, m2
ηI
m2
Nk
:
→ (Mν)ij =
λ5v2
8π2
k
hijhjk
mNk
log
m2
Nk
m2
0
− 1 ,
A4 assignment:
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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43. Radiative Seesaw with A4
Minimal scotogenic model with ∆m2
η m2
ηR
, m2
ηI
m2
Nk
:
→ (Mν)ij =
λ5v2
8π2
k
hijhjk
mNk
log
m2
Nk
m2
0
− 1 ,
A4 assignment: Nk ∼ 3,
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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44. Radiative Seesaw with A4
Minimal scotogenic model with ∆m2
η m2
ηR
, m2
ηI
m2
Nk
:
→ (Mν)ij =
λ5v2
8π2
k
hijhjk
mNk
log
m2
Nk
m2
0
− 1 ,
A4 assignment: Nk ∼ 3, η ∼ 1,
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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45. Radiative Seesaw with A4
Minimal scotogenic model with ∆m2
η m2
ηR
, m2
ηI
m2
Nk
:
→ (Mν)ij =
λ5v2
8π2
k
hijhjk
mNk
log
m2
Nk
m2
0
− 1 ,
A4 assignment: Nk ∼ 3, η ∼ 1, (νi, li)L ∼ 3,
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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46. Radiative Seesaw with A4
Minimal scotogenic model with ∆m2
η m2
ηR
, m2
ηI
m2
Nk
:
→ (Mν)ij =
λ5v2
8π2
k
hijhjk
mNk
log
m2
Nk
m2
0
− 1 ,
A4 assignment: Nk ∼ 3, η ∼ 1, (νi, li)L ∼ 3, liR ∼ 1, 1 , 1
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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47. Radiative Seesaw with A4
Minimal scotogenic model with ∆m2
η m2
ηR
, m2
ηI
m2
Nk
:
→ (Mν)ij =
λ5v2
8π2
k
hijhjk
mNk
log
m2
Nk
m2
0
− 1 ,
A4 assignment: Nk ∼ 3, η ∼ 1, (νi, li)L ∼ 3, liR ∼ 1, 1 , 1
MN =
A F E
F A D
E D A
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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48. Radiative Seesaw with A4
Minimal scotogenic model with ∆m2
η m2
ηR
, m2
ηI
m2
Nk
:
→ (Mν)ij =
λ5v2
8π2
k
hijhjk
mNk
log
m2
Nk
m2
0
− 1 ,
A4 assignment: Nk ∼ 3, η ∼ 1, (νi, li)L ∼ 3, liR ∼ 1, 1 , 1
MN =
A F E
F A D
E D A
Rotate to TBM basis, assume F = −E (via interchange symmetry)
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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49. Radiative Seesaw with A4
Parameters to describe neutrino oscillation :
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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50. Radiative Seesaw with A4
Parameters to describe neutrino oscillation : A, CR, CI, DR, DI.
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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51. Radiative Seesaw with A4
Parameters to describe neutrino oscillation : A, CR, CI, DR, DI.
Goals:
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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52. Radiative Seesaw with A4
Parameters to describe neutrino oscillation : A, CR, CI, DR, DI.
Goals: constrain parameters from ∆m2
ij’s, and θ12, θ23 → predict θ13,
mee, and δCP .
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
11/39
53. Radiative Seesaw with A4
Parameters to describe neutrino oscillation : A, CR, CI, DR, DI.
Goals: constrain parameters from ∆m2
ij’s, and θ12, θ23 → predict θ13,
mee, and δCP .
Challenge:
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
11/39
54. Radiative Seesaw with A4
Parameters to describe neutrino oscillation : A, CR, CI, DR, DI.
Goals: constrain parameters from ∆m2
ij’s, and θ12, θ23 → predict θ13,
mee, and δCP .
Challenge:
mν =
h2
mNk
log(m2
Nk
/m2
0) − 1
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
11/39
55. Radiative Seesaw with A4
Parameters to describe neutrino oscillation : A, CR, CI, DR, DI.
Goals: constrain parameters from ∆m2
ij’s, and θ12, θ23 → predict θ13,
mee, and δCP .
Challenge:
mν =
h2
mNk
log(m2
Nk
/m2
0) − 1
→ non-linear relationship between ∆m2
ij and mass-matrix parameters
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
11/39
56. Radiative Seesaw with A4
Parameters to describe neutrino oscillation : A, CR, CI, DR, DI.
Goals: constrain parameters from ∆m2
ij’s, and θ12, θ23 → predict θ13,
mee, and δCP .
Challenge:
mν =
h2
mNk
log(m2
Nk
/m2
0) − 1
→ non-linear relationship between ∆m2
ij and mass-matrix parameters
Solution: log(m2
Nk
/m2
0) adds a comparatively small correction to seesaw
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
11/39
57. Radiative Seesaw with A4
Parameters to describe neutrino oscillation : A, CR, CI, DR, DI.
Goals: constrain parameters from ∆m2
ij’s, and θ12, θ23 → predict θ13,
mee, and δCP .
Challenge:
mν =
h2
mNk
log(m2
Nk
/m2
0) − 1
→ non-linear relationship between ∆m2
ij and mass-matrix parameters
Solution: log(m2
Nk
/m2
0) adds a comparatively small correction to seesaw
→ scan for exact solutions and extract parameters, and then use
Mathematica’s FindRoot function to produce numerical solutions for the
full radiative seesaw
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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58. Radiative Seesaw with A4
0.06 0.08 0.10 0.12 0.14
0.850
0.855
0.860
0.865
0.870
0.875
sin2
2Θ13
sin2
2Θ12
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
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59. Radiative Seesaw with A4
0.06 0.08 0.10 0.12 0.14
1
2
3
4
5
sin2
2Θ13
tan∆CP
QD, Im D Re D
QD, Im D 0
NH, Im D 0
IH, Im D Re D
IH, Im D 0
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
13/39
60. Scotogenic ∆(27) Models
∆(27) instead of A4 as flavor symmetry has interesting properties:
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
14/39
61. Scotogenic ∆(27) Models
∆(27) instead of A4 as flavor symmetry has interesting properties:
Naturally predicts θ13 = 0 before discovery
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
14/39
62. Scotogenic ∆(27) Models
∆(27) instead of A4 as flavor symmetry has interesting properties:
Naturally predicts θ13 = 0 before discovery
∆(27) can lead to spontaneous CP violation
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
14/39
63. Scotogenic ∆(27) Models
∆(27) instead of A4 as flavor symmetry has interesting properties:
Naturally predicts θ13 = 0 before discovery
∆(27) can lead to spontaneous CP violation
Leads to special form of ν mass matrix using:
Mν =
fa c b
c fb a
b a fc
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
14/39
64. Scotogenic ∆(27) Models
∆(27) instead of A4 as flavor symmetry has interesting properties:
Naturally predicts θ13 = 0 before discovery
∆(27) can lead to spontaneous CP violation
Leads to special form of ν mass matrix using:
Mν =
fa c b
c fb a
b a fc
Two neutrino oscillation patterns: Ml diagonal and non-diagonal.
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
14/39
65. Scotogenic ∆(27) Models
∆(27) instead of A4 as flavor symmetry has interesting properties:
Naturally predicts θ13 = 0 before discovery
∆(27) can lead to spontaneous CP violation
Leads to special form of ν mass matrix using:
Mν =
fa c b
c fb a
b a fc
Two neutrino oscillation patterns: Ml diagonal and non-diagonal.
Both previously studied in a non-scotogenic context.
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
14/39
66. Scotogenic ∆(27) Models
General procedure:
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
15/39
67. Scotogenic ∆(27) Models
General procedure:
Use approx. solutions as starting place for full parameter scan
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
15/39
68. Scotogenic ∆(27) Models
General procedure:
Use approx. solutions as starting place for full parameter scan
Numerically diagonalize Mν
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
15/39
69. Scotogenic ∆(27) Models
General procedure:
Use approx. solutions as starting place for full parameter scan
Numerically diagonalize Mν
Start with κ real and κ imaginary, then arbitrary κ phase
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
15/39
70. Scotogenic ∆(27) Models
General procedure:
Use approx. solutions as starting place for full parameter scan
Numerically diagonalize Mν
Start with κ real and κ imaginary, then arbitrary κ phase
Extract CP phase with Jarlskog invariant:
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
15/39
71. Scotogenic ∆(27) Models
General procedure:
Use approx. solutions as starting place for full parameter scan
Numerically diagonalize Mν
Start with κ real and κ imaginary, then arbitrary κ phase
Extract CP phase with Jarlskog invariant:
JCP = Im(Uµ3U∗
e3Ue2U∗
µ2)
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
15/39
72. Scotogenic ∆(27) Models
General procedure:
Use approx. solutions as starting place for full parameter scan
Numerically diagonalize Mν
Start with κ real and κ imaginary, then arbitrary κ phase
Extract CP phase with Jarlskog invariant:
JCP = Im(Uµ3U∗
e3Ue2U∗
µ2)
Yields predictions for δCP , θ23, mee
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
15/39
73. Scotogenic ∆(27) Models
General procedure:
Use approx. solutions as starting place for full parameter scan
Numerically diagonalize Mν
Start with κ real and κ imaginary, then arbitrary κ phase
Extract CP phase with Jarlskog invariant:
JCP = Im(Uµ3U∗
e3Ue2U∗
µ2)
Yields predictions for δCP , θ23, mee
→ Ml non-diagonal case yields essentially same results
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
15/39
74. Scotogenic ∆(27) Models
Κ Re
Κ Im
0.84 0.85 0.86 0.87
0.03
0.04
0.05
0.06
sin2
2Θ12
mee
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
16/39
75. Scotogenic ∆(27) Models
0.085 0.090 0.095 0.100 0.105
0.0320
0.0325
0.0330
0.0335
0.0340
0.0345
0.0350
0.0355
0.0360
sin2
2Θ13
Jcp
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
17/39
76. µ − τ Interchange Symmetry
ν νs
N N
×
E0
E0
φ0
φ0
Figure 1 : One-loop generation of inverse seesaw neutrino mass.
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
18/39
77. µ − τ Interchange Symmetry
Mass matrix from invariant Dirac masses and N − E mixing:
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
19/39
78. µ − τ Interchange Symmetry
Mass matrix from invariant Dirac masses and N − E mixing:
MN,E =
mN mD
mF mE
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
19/39
79. µ − τ Interchange Symmetry
Mass matrix from invariant Dirac masses and N − E mixing:
MN,E =
mN mD
mF mE
Diagonalized via:
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
19/39
80. µ − τ Interchange Symmetry
Mass matrix from invariant Dirac masses and N − E mixing:
MN,E =
mN mD
mF mE
Diagonalized via:
mDmE + mF mN = sin θL cos θL(m2
1 − m2
2)
mDmN + mF mE = sin θR cos θR(m2
1 − m2
2)
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
19/39
81. µ − τ Interchange Symmetry
Mass matrix from invariant Dirac masses and N − E mixing:
MN,E =
mN mD
mF mE
Diagonalized via:
mDmE + mF mN = sin θL cos θL(m2
1 − m2
2)
mDmN + mF mE = sin θR cos θR(m2
1 − m2
2)
In terms of mass eigenstates:
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
19/39
82. µ − τ Interchange Symmetry
Mass matrix from invariant Dirac masses and N − E mixing:
MN,E =
mN mD
mF mE
Diagonalized via:
mDmE + mF mN = sin θL cos θL(m2
1 − m2
2)
mDmN + mF mE = sin θR cos θR(m2
1 − m2
2)
In terms of mass eigenstates:
ω1
(R,L)
ω2
(R,L)
=
cos θ(R,L) − sin θ(R,L)
sin θ(R,L) cos θ(R,L)
N(R,L)
E(R,L)
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
19/39
83. µ − τ Interchange Symmetry
mν calculated via:
ν ν
s
mR
2
× ×
| |
(cR ¯ωc
1, sR ¯ωc
2) (cRωc
1, sRωc
2)
(cRωc
2,−sRωc
1) (cR ¯ω2,−sR ¯ω1)
ν ν
s
mL
2
× ×
| |
(cL ¯ωc
1, sL ¯ωc
2) (cLωc
1, sLωc
2)
(cRωc
2,−sRωc
1) (cR ¯ω2,−sR ¯ω1)
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
20/39
84. µ − τ Interchange Symmetry
mν calculated via:
ν ν
s
mR
2
× ×
| |
(cR ¯ωc
1, sR ¯ωc
2) (cRωc
1, sRωc
2)
(cRωc
2,−sRωc
1) (cR ¯ω2,−sR ¯ω1)
ν ν
s
mL
2
× ×
| |
(cL ¯ωc
1, sL ¯ωc
2) (cLωc
1, sLωc
2)
(cRωc
2,−sRωc
1) (cR ¯ω2,−sR ¯ω1)
Previously studied with λ = fµ/fτ = 1, but not strictly necessary.
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
20/39
85. µ − τ Interchange Symmetry
mν calculated via:
ν ν
s
mR
2
× ×
| |
(cR ¯ωc
1, sR ¯ωc
2) (cRωc
1, sRωc
2)
(cRωc
2,−sRωc
1) (cR ¯ω2,−sR ¯ω1)
ν ν
s
mL
2
× ×
| |
(cL ¯ωc
1, sL ¯ωc
2) (cLωc
1, sLωc
2)
(cRωc
2,−sRωc
1) (cR ¯ω2,−sR ¯ω1)
Previously studied with λ = fµ/fτ = 1, but not strictly necessary.
When λ = 1: δcp = ±π
2 and θ23 is maximal.
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
20/39
86. µ − τ Interchange Symmetry
mν calculated via:
ν ν
s
mR
2
× ×
| |
(cR ¯ωc
1, sR ¯ωc
2) (cRωc
1, sRωc
2)
(cRωc
2,−sRωc
1) (cR ¯ω2,−sR ¯ω1)
ν ν
s
mL
2
× ×
| |
(cL ¯ωc
1, sL ¯ωc
2) (cLωc
1, sLωc
2)
(cRωc
2,−sRωc
1) (cR ¯ω2,−sR ¯ω1)
Previously studied with λ = fµ/fτ = 1, but not strictly necessary.
When λ = 1: δcp = ±π
2 and θ23 is maximal.
Full numerical solutions studied w/ λ = 1
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
20/39
87. µ − τ Interchange Symmetry
Λ 1.15 1.08 1.04
m1 eV
0
0.03
0.06
0.94 0.95 0.96 0.97 0.98 0.99 1.00
0.980
0.985
0.990
0.995
1.000
∆cp
Π 2
sin2
2Θ23
Figure 2 : Normal Hierarchy
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
21/39
88. µ − τ Interchange Symmetry
Λ 0.87 0.92 0.96
m1 eV
0
0.03
0.06
0.990 0.992 0.994 0.996 0.998 1.000
0.980
0.985
0.990
0.995
1.000
∆cp
Π 2
sin2
2Θ23
Figure 3 : Normal Hierarchy for λ → λ−1
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
22/39
89. µ − τ Interchange Symmetry
Λ 1.16 1.08 1.03
m3 eV
0
0.03
0.06
0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
0.980
0.985
0.990
0.995
1.000
∆cp
Π 2
sin2
2Θ23
Figure 4 : Inverted Hierarchy
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
23/39
90. Scotogenic WDM
Scotogenic model also has rich collider phenomenology.
Assume m2
Nk
m2
R, m2
I, yields:
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
24/39
91. Scotogenic WDM
Scotogenic model also has rich collider phenomenology.
Assume m2
Nk
m2
R, m2
I, yields:
Mν
ij =
k
hijhjk
log(m2
R/m2
I)mNk
16π2
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
24/39
92. Scotogenic WDM
Scotogenic model also has rich collider phenomenology.
Assume m2
Nk
m2
R, m2
I, yields:
Mν
ij =
k
hijhjk
log(m2
R/m2
I)mNk
16π2
mν ∝ h2mN , even with h 1 mN must be small
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
24/39
93. Scotogenic WDM
Scotogenic model also has rich collider phenomenology.
Assume m2
Nk
m2
R, m2
I, yields:
Mν
ij =
k
hijhjk
log(m2
R/m2
I)mNk
16π2
mν ∝ h2mN , even with h 1 mN must be small
No assumptions about mR/mI, but splitting as large as 110 GeV:
log(m2
R/m2
I) ≈ 0.94
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
24/39
94. Scotogenic WDM
Scotogenic model also has rich collider phenomenology.
Assume m2
Nk
m2
R, m2
I, yields:
Mν
ij =
k
hijhjk
log(m2
R/m2
I)mNk
16π2
mν ∝ h2mN , even with h 1 mN must be small
No assumptions about mR/mI, but splitting as large as 110 GeV:
log(m2
R/m2
I) ≈ 0.94
Assign L = −1 to Nks → mN violates L by 2, becomes (−1)L
naturally leads to mN = O(10) keV
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
24/39
95. Scotogenic WDM
Scotogenic model also has rich collider phenomenology.
Assume m2
Nk
m2
R, m2
I, yields:
Mν
ij =
k
hijhjk
log(m2
R/m2
I)mNk
16π2
mν ∝ h2mN , even with h 1 mN must be small
No assumptions about mR/mI, but splitting as large as 110 GeV:
log(m2
R/m2
I) ≈ 0.94
Assign L = −1 to Nks → mN violates L by 2, becomes (−1)L
naturally leads to mN = O(10) keV
Warm Dark Matter Candidate
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
24/39
96. Scotogenic WDM
Lightest Nk is absolutely stable
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
25/39
97. Scotogenic WDM
Lightest Nk is absolutely stable → no contribution to x-ray emissions.
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
25/39
98. Scotogenic WDM
Lightest Nk is absolutely stable → no contribution to x-ray emissions.
Problem: mN adds to FCNC, µ magnetic moment, etc.
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
25/39
99. Scotogenic WDM
Lightest Nk is absolutely stable → no contribution to x-ray emissions.
Problem: mN adds to FCNC, µ magnetic moment, etc.
Solution: A4 flavor symmetry suppresses at leading order:
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
25/39
100. Scotogenic WDM
Lightest Nk is absolutely stable → no contribution to x-ray emissions.
Problem: mN adds to FCNC, µ magnetic moment, etc.
Solution: A4 flavor symmetry suppresses at leading order:
η ∼ 1, (νi, li), Nk ∼ 3 → hik = hδik
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
25/39
102. Scotogenic WDM
mN ∼ 10 keV allows η± → e±N1, µ±N2, potential production at LHC
η pair-produced by Z/γ → similar to pair-produced W’s
:
mη± (GeV) σ (fb) w/ Emiss
T cuts (GeV): 0 25 50 100
80 33.2 27.9 18.3 2.88
90 22.7 19.8 14.4 3.10
100 15.7 14.0 10.6 3.08
110 11.4 10.3 8.13 3.03
120 8.72 7.99 6.54 2.91
130 6.45 5.98 5.05 2.57
140 4.97 4.64 3.96 2.21
150 3.84 3.62 3.16 1.89
SM Background σ (fb): 626 453 206 8.60
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
26/39
103. Scotogenic WDM
mN ∼ 10 keV allows η± → e±N1, µ±N2, potential production at LHC
η pair-produced by Z/γ → similar to pair-produced W’s
Too similar of a signature to W background to be found at 7,8 TeV LHC:
mη± (GeV) σ (fb) w/ Emiss
T cuts (GeV): 0 25 50 100
80 33.2 27.9 18.3 2.88
90 22.7 19.8 14.4 3.10
100 15.7 14.0 10.6 3.08
110 11.4 10.3 8.13 3.03
120 8.72 7.99 6.54 2.91
130 6.45 5.98 5.05 2.57
140 4.97 4.64 3.96 2.21
150 3.84 3.62 3.16 1.89
SM Background σ (fb): 626 453 206 8.60
S. Bhattacharya, E. Ma, A. Natale, and A. Rashed, Phys. Rev. D87 (2013) 097301, arXiv:1302.6266 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
26/39
104. Scotogenic Color-Triplets
×
uL uRNR NL
ξ2/3
ζ2/3
φ0
×
dL dRNR NL
ξ−1/3
ζ−1/3
φ0
×
lL lRNR NL
η+
χ+
φ0
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
27/39
105. Scotogenic Color-Triplets
Usual minimal scotogenic content: η, Ni (now a Dirac fermion)
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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106. Scotogenic Color-Triplets
Usual minimal scotogenic content: η, Ni (now a Dirac fermion)
Non-minimal particles:
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
28/39
107. Scotogenic Color-Triplets
Usual minimal scotogenic content: η, Ni (now a Dirac fermion)
Non-minimal particles:
χ+ singlet, scalar
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
28/39
108. Scotogenic Color-Triplets
Usual minimal scotogenic content: η, Ni (now a Dirac fermion)
Non-minimal particles:
χ+ singlet, scalar
(ξ2/3, ξ−1/3) color-triplet, SU(2) doublet, scalar
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
28/39
109. Scotogenic Color-Triplets
Usual minimal scotogenic content: η, Ni (now a Dirac fermion)
Non-minimal particles:
χ+ singlet, scalar
(ξ2/3, ξ−1/3) color-triplet, SU(2) doublet, scalar
ζ−1/3 color-triplet, singlet, scalar
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
28/39
110. Scotogenic Color-Triplets
Usual minimal scotogenic content: η, Ni (now a Dirac fermion)
Non-minimal particles:
χ+ singlet, scalar
(ξ2/3, ξ−1/3) color-triplet, SU(2) doublet, scalar
ζ−1/3 color-triplet, singlet, scalar
New Yukawa interactions:
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
28/39
111. Scotogenic Color-Triplets
Usual minimal scotogenic content: η, Ni (now a Dirac fermion)
Non-minimal particles:
χ+ singlet, scalar
(ξ2/3, ξ−1/3) color-triplet, SU(2) doublet, scalar
ζ−1/3 color-triplet, singlet, scalar
New Yukawa interactions:
L = f( ¯dRN1L + ¯sRN2L)ζ−1/3
+ f (¯eRN1L + ¯µRN2L)χ−
+ h.c.,
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
28/39
112. Scotogenic Color-Triplets
Usual minimal scotogenic content: η, Ni (now a Dirac fermion)
Non-minimal particles:
χ+ singlet, scalar
(ξ2/3, ξ−1/3) color-triplet, SU(2) doublet, scalar
ζ−1/3 color-triplet, singlet, scalar
New Yukawa interactions:
L = f( ¯dRN1L + ¯sRN2L)ζ−1/3
+ f (¯eRN1L + ¯µRN2L)χ−
+ h.c.,
where mζ > mN2 > mχ > mN1
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
28/39
113. Scotogenic Color-Triplets
Usual minimal scotogenic content: η, Ni (now a Dirac fermion)
Non-minimal particles:
χ+ singlet, scalar
(ξ2/3, ξ−1/3) color-triplet, SU(2) doublet, scalar
ζ−1/3 color-triplet, singlet, scalar
New Yukawa interactions:
L = f( ¯dRN1L + ¯sRN2L)ζ−1/3
+ f (¯eRN1L + ¯µRN2L)χ−
+ h.c.,
where mζ > mN2 > mχ > mN1
ζ → dN1, sN2, and N2 → eµN1 via χ+
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
28/39
114. Scotogenic Color-Triplets
Complementarity:
ζ−1/3
N1
d
N1
d
relic abundance and direct detection are important constraints
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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115. Scotogenic Color-Triplets
Complementarity:
ζ−1/3
N1
d
N1
d
Direct Detection
relic abundance and direct detection are important constraints
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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117. Scotogenic Color-Triplets
Relic density for DM with color-triplet, scalar, mediator previously
calculated (see arXiv:1308.0612):
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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118. Scotogenic Color-Triplets
Relic density for DM with color-triplet, scalar, mediator previously
calculated (see arXiv:1308.0612):
σv =
1
2
3f4m2
N1
32π(m2
N1
+ m2
ζ)2
+ v2
f4m2
N1
(11m4
ζ − 5m4
N1
− 18m2
N1
m2
ζ
256π(m2
N1
+ m2
ζ)4
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
30/39
119. Scotogenic Color-Triplets
Relic density for DM with color-triplet, scalar, mediator previously
calculated (see arXiv:1308.0612):
σv =
1
2
3f4m2
N1
32π(m2
N1
+ m2
ζ)2
+ v2
f4m2
N1
(11m4
ζ − 5m4
N1
− 18m2
N1
m2
ζ
256π(m2
N1
+ m2
ζ)4
→ with mass choices cannot fit Ω0h2 for DM unless f > 0.5, however f
also needed to radiatively generate mu. To get correct mu values
f ≈ 0.01.
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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120. Scotogenic Color-Triplets
Relic density for DM with color-triplet, scalar, mediator previously
calculated (see arXiv:1308.0612):
σv =
1
2
3f4m2
N1
32π(m2
N1
+ m2
ζ)2
+ v2
f4m2
N1
(11m4
ζ − 5m4
N1
− 18m2
N1
m2
ζ
256π(m2
N1
+ m2
ζ)4
→ with mass choices cannot fit Ω0h2 for DM unless f > 0.5, however f
also needed to radiatively generate mu. To get correct mu values
f ≈ 0.01.
Solution:
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
30/39
121. Scotogenic Color-Triplets
Relic density for DM with color-triplet, scalar, mediator previously
calculated (see arXiv:1308.0612):
σv =
1
2
3f4m2
N1
32π(m2
N1
+ m2
ζ)2
+ v2
f4m2
N1
(11m4
ζ − 5m4
N1
− 18m2
N1
m2
ζ
256π(m2
N1
+ m2
ζ)4
→ with mass choices cannot fit Ω0h2 for DM unless f > 0.5, however f
also needed to radiatively generate mu. To get correct mu values
f ≈ 0.01.
Solution: N1N1 → e+e−, using MicrOMEGAs with f ≈ 0.5, f ≈ 0.01,
yields correct Ω0h2. f is partially constrained by ml
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
30/39
122. Scotogenic Color-Triplets
Relic density for DM with color-triplet, scalar, mediator previously
calculated (see arXiv:1308.0612):
σv =
1
2
3f4m2
N1
32π(m2
N1
+ m2
ζ)2
+ v2
f4m2
N1
(11m4
ζ − 5m4
N1
− 18m2
N1
m2
ζ
256π(m2
N1
+ m2
ζ)4
→ with mass choices cannot fit Ω0h2 for DM unless f > 0.5, however f
also needed to radiatively generate mu. To get correct mu values
f ≈ 0.01.
Solution: N1N1 → e+e−, using MicrOMEGAs with f ≈ 0.5, f ≈ 0.01,
yields correct Ω0h2. f is partially constrained by ml → possible
consequences for η Yukawa couplings to produce observed ml.
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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126. Scotogenic Color-Triplets
Main collider production:
g
g
ζ−1/3
ζ−1/3
g
g
g
ζ−1/3
ζ−1/3
Other diagrams include quarks, t-channel, etc. but dominated by gg → ζζ
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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127. Scotogenic Color-Triplets
L = f( ¯dRN1L + ¯sRN2L)ζ−1/3
+ f (¯eRN1L + ¯µRN2L)χ−
+ h.c.
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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128. Scotogenic Color-Triplets
L = f( ¯dRN1L + ¯sRN2L)ζ−1/3
+ f (¯eRN1L + ¯µRN2L)χ−
+ h.c.
Generic Signatures:
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
33/39
129. Scotogenic Color-Triplets
L = f( ¯dRN1L + ¯sRN2L)ζ−1/3
+ f (¯eRN1L + ¯µRN2L)χ−
+ h.c.
Generic Signatures:
2 jets + Emiss
T
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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130. Scotogenic Color-Triplets
L = f( ¯dRN1L + ¯sRN2L)ζ−1/3
+ f (¯eRN1L + ¯µRN2L)χ−
+ h.c.
Generic Signatures:
2 jets + Emiss
T ← same as SUSY searches
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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131. Scotogenic Color-Triplets
L = f( ¯dRN1L + ¯sRN2L)ζ−1/3
+ f (¯eRN1L + ¯µRN2L)χ−
+ h.c.
Generic Signatures:
2 jets + Emiss
T ← same as SUSY searches
2 jets + 2 leptons (opposite-sign opposite-flavor) + Emiss
T
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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132. Scotogenic Color-Triplets
L = f( ¯dRN1L + ¯sRN2L)ζ−1/3
+ f (¯eRN1L + ¯µRN2L)χ−
+ h.c.
Generic Signatures:
2 jets + Emiss
T ← same as SUSY searches
2 jets + 2 leptons (opposite-sign opposite-flavor) + Emiss
T
2 jets + four leptons + Emiss
T
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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133. Scotogenic Color-Triplets
L = f( ¯dRN1L + ¯sRN2L)ζ−1/3
+ f (¯eRN1L + ¯µRN2L)χ−
+ h.c.
Generic Signatures:
2 jets + Emiss
T ← same as SUSY searches
2 jets + 2 leptons (opposite-sign opposite-flavor) + Emiss
T
2 jets + four leptons + Emiss
T
A common SUSY signature is opposite-sign same-flavor, positive signals
(even < 5σ) yield constrains/rule out this scotogenic model (OSOF
estimates flavor symmetric background so signal constrains excess OSOF)
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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134. Scotogenic Color-Triplets
Simplified SUSY topologies: gluino & squarks decouple, simplifying
assumptions about squark-quark coupling (CMS-PAS-SUS-13-019):
(GeV)qm
400 600 800 1000 1200
(GeV)LSPm
0
200
400
600
800
-2
10
-1
10
1
10
= 8 TeVs,-1
CMS Preliminary, 19.5 fb
NLO+NLL exclusion
1
0
χ∼q→q~,q~q~→pp
)c~,s~,d
~
,u~(
R
q~+
L
q~
q~one light
theory
σ1±Observed
experimentσ1±Expected
95%C.L.upperlimitoncrosssection(pb)
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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135. Scotogenic Color-Triplets
Simplified SUSY topologies: gluino & squarks decouple, simplifying
assumptions about squark-quark coupling (CMS-PAS-SUS-13-019):
(GeV)qm
400 600 800 1000 1200
(GeV)LSPm
0
200
400
600
800
-2
10
-1
10
1
10
= 8 TeVs,-1
CMS Preliminary, 19.5 fb
NLO+NLL exclusion
1
0
χ∼q→q~,q~q~→pp
)c~,s~,d
~
,u~(
R
q~+
L
q~
q~one light
theory
σ1±Observed
experimentσ1±Expected
95%C.L.upperlimitoncrosssection(pb)
→ mζ > 400 GeV
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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136. Scotogenic Color-Triplets
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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137. Scotogenic Color-Triplets
SUSY Simplified topology production compared to LO calculation in
CalcHEP to scale to NLO values.
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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138. Scotogenic Color-Triplets
SUSY Simplified topology production compared to LO calculation in
CalcHEP to scale to NLO values.
Main background from ¯tt decays, calculated & scaled to NLO.
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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139. Scotogenic Color-Triplets
SUSY Simplified topology production compared to LO calculation in
CalcHEP to scale to NLO values.
Main background from ¯tt decays, calculated & scaled to NLO.
Various cuts and masses tried, final analysis uses mN2 = 400 GeV,
mχ = 200 GeV, mζ > 400 GeV, 180 GeV ≥ mN1 ≥ 100 GeV
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
35/39
140. Scotogenic Color-Triplets
SUSY Simplified topology production compared to LO calculation in
CalcHEP to scale to NLO values.
Main background from ¯tt decays, calculated & scaled to NLO.
Various cuts and masses tried, final analysis uses mN2 = 400 GeV,
mχ = 200 GeV, mζ > 400 GeV, 180 GeV ≥ mN1 ≥ 100 GeV
6 cut regions used, only 4 cuts produce large enough
signal-to-background
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
35/39
141. Scotogenic Color-Triplets
SUSY Simplified topology production compared to LO calculation in
CalcHEP to scale to NLO values.
Main background from ¯tt decays, calculated & scaled to NLO.
Various cuts and masses tried, final analysis uses mN2 = 400 GeV,
mχ = 200 GeV, mζ > 400 GeV, 180 GeV ≥ mN1 ≥ 100 GeV
6 cut regions used, only 4 cuts produce large enough
signal-to-background
Cuts on scalar sum of hadronic transverse momentum (HT ) utilized
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
35/39
142. Scotogenic Color-Triplets
SUSY Simplified topology production compared to LO calculation in
CalcHEP to scale to NLO values.
Main background from ¯tt decays, calculated & scaled to NLO.
Various cuts and masses tried, final analysis uses mN2 = 400 GeV,
mχ = 200 GeV, mζ > 400 GeV, 180 GeV ≥ mN1 ≥ 100 GeV
6 cut regions used, only 4 cuts produce large enough
signal-to-background
Cuts on scalar sum of hadronic transverse momentum (HT ) utilized
For simulation: ECM = 13 TeV, CTEQ6M, CalcHEP → PYTHIA 8
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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143. Scotogenic Color-Triplets
400 500 600 700 800 900
100
120
140
160
180
mΖ GeV
mN1
GeV
R2
R3
R2 Cut: Emiss
T > 200 GeV, HT > 600 GeV, p
jet
T
> 30 GeV, p
lep
T
> 20 GeV, σSM = 10.0 fb
R3 Cut: Emiss
T > 275 GeV, HT > 600 GeV, p
jet
T
> 30 GeV, p
lep
T
> 20 GeV, σSM = 0.4 fb
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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144. Scotogenic Color-Triplets
400 500 600 700 800 900
100
120
140
160
180
mΖ GeV
mN1
GeV
R5
R6
R5 Cut: Emiss
T > 200 GeV, HT > 350 GeV, p
jet
T
> 30 GeV, p
lep
T
> 20 GeV, σSM = 7.1 fb
R6 Cut: Emiss
T > 200 GeV, HT > 350 GeV, p
jet
T
> 150 GeV, p
lep
T
> 25 GeV, σSM = 1.2 fb
E. Ma, and A. Natale, Phys. Lett. B740 (2015) 80-82, arXiv:1410.2902 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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145. Conclusion & Discussion
Scotogenic model has rich phenomenology
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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146. Conclusion & Discussion
Scotogenic model has rich phenomenology
Discrete flavor symmetries still promising, particularly when used with
the scotogenic mechanism
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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147. Conclusion & Discussion
Scotogenic model has rich phenomenology
Discrete flavor symmetries still promising, particularly when used with
the scotogenic mechanism
Extensions to minimal scotogenic model potentially discoverable at 13
TeV LHC & distinct from SUSY signatures
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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148. Conclusion & Discussion
Scotogenic model has rich phenomenology
Discrete flavor symmetries still promising, particularly when used with
the scotogenic mechanism
Extensions to minimal scotogenic model potentially discoverable at 13
TeV LHC & distinct from SUSY signatures
Modified DM phenomenology potential for future study:
multi-partite, WDM, SIMPs, CDM, etc.
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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150. Dark Matter
Evidence for Dark Matter:
Rotation curves of spiral galaxies
Strong and weak gravitational lensing
Cosmic Microwave Background (CMB) fits
Bullet Cluster
Baryonic Acoustic Oscillations (BAO)
Galaxy formation
Large scale structure simulation
Mass to luminosity ratios
See arXiv:1001.0316 for a review on DM evidence.
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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151. Neutrino Mass in the SM
If we write Dirac’s equation in the chiral representation (ψL = 1−γ5
2 Ψ,
ψR = 1+γ5
2 Ψ), then we get:
L = ψL/pψL + ψR/pψR + m(ψRψL + ψLψR)
No right-handed neutrino has been observed experimentally (any
right-handed ν would only couple to neutrinos to explain observations)
o In the minimal Standard Model, there is no neutrino mass term.
o Neutrinos have been observed to oscillate, which implies a non-zero
neutrino mass.
→ Tension between theory and experiment.
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
152. Neutrino Oscillation and Mass
Suppose a neutrino is created in a single flavor state (ie e, µ, τ) and
travels as a plane wave to our detector (distance L in time T). If it is in a
superposition of mass eigenstates, νl = n Cl nνn, then:
νl(t) =
n
Cl ne−i(Ent−pnx)
νn
The transition amplitude, νl | νl(t) , is:
A(l → l )(x, t) =
n
C†
l nCl ne−i(Ent−pnx)
Squaring this yields the probability of finding the neutrino in a new flavor
state (P(l → l ) for l = l ):
n
|Cl n|2
|Cl n|2
+2
m>n
|C†
l mCl mC†
l nCl n| cos(∆Em,nt−∆pm,nx+φl lmn)
φl lmn = Arg(C†
l mCl mC†
l nCl n)
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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153. Neutrino Oscillation: Definition
So our oscillation probability is:
n
|Cl n|2
|Cl n|2
+2
m>n
|C†
l mCl mC†
l nCl n| cos(∆Em,nt−∆pm,nx+φl lmn)
φl lmn = Arg(C†
l mCl mC†
l nCl n)
Assume, for simplicity, that the wave packet travels with the average
energy and momentum of the mass eigenstates:
¯v =
Em
Em + En
vm +
En
Em + En
vn, with vm =
pm
Em
∆Em,nT−∆pm,nL = (
∆Em,n
v
−∆pm,n)L = (
∆Em,n(Em + En)
Emvm + Envn
−∆pm,n)L
=
E2
m − E2
n − (p2
m − p2
n)
pm + pn
L = −
(m2
m − m2
n)
pm + pn
L
If m2
m − m2
n = 0, as the minimal Standard Model indicates, there
should be no neutrino oscillations (ie P(l → l ) = 0 if l = l ).
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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154. Flavor Symmetries: Introduction to A4
Symmetry of the tetrahedron:→ Alternating group of 4 elements
o Smallest non-Abelian discrete symmetry with 3-dim representation
o Multiplication rules: 3 × 3 = 1 + 1’ + 1” + 3 + 3
1’ × 1’ = 1”
1” × 1” = 1’
1’ × 1” = 1
Writing the products explicitly (arXiv:1003.3552) (A)3 × (B)3:
→ (AxBx + AyBy + AzBz)1
→ (AxBx + ωAyBy + ω2
AzBz)1’
→ (AxBx + ω2
AyBy + ωAzBz)1”
ω = ei 2π
3 , and ω2
= ω
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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155. Backup Slides: Radiative Seesaw with A4
MTBM
N =
A + D 0 0
0 A C
0 C A − D
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
156. Backup Slides: Radiative Seesaw with A4
MTBM
N =
A + D 0 0
0 A C
0 C A − D
Physical mixing angles are extracted by diagonalizing MN
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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157. Backup Slides: Radiative Seesaw with A4
MTBM
N =
A + D 0 0
0 A C
0 C A − D
Physical mixing angles are extracted by diagonalizing MN
mass eigenvalues are of the form of seesaw with additional log term from
loop (radiative seesaw).
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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158. Backup Slides: Radiative Seesaw with A4
MTBM
N =
A + D 0 0
0 A C
0 C A − D
Physical mixing angles are extracted by diagonalizing MN
mass eigenvalues are of the form of seesaw with additional log term from
loop (radiative seesaw).
Diagonalize MTBM
N with rotation matrix:
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
159. Backup Slides: Radiative Seesaw with A4
MTBM
N =
A + D 0 0
0 A C
0 C A − D
Physical mixing angles are extracted by diagonalizing MN
mass eigenvalues are of the form of seesaw with additional log term from
loop (radiative seesaw).
Diagonalize MTBM
N with rotation matrix:
tan 2θ =
2(4A2C2
R − 4ACR(CRDR + CIDI) + (C2
R + C2
I )(D2
R + D2
I ))1/2
2ADR − (D2
R + D2
I )
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
160. Backup Slides: Radiative Seesaw with A4
MTBM
N =
A + D 0 0
0 A C
0 C A − D
Physical mixing angles are extracted by diagonalizing MN
mass eigenvalues are of the form of seesaw with additional log term from
loop (radiative seesaw).
Diagonalize MTBM
N with rotation matrix:
tan 2θ =
2(4A2C2
R − 4ACR(CRDR + CIDI) + (C2
R + C2
I )(D2
R + D2
I ))1/2
2ADR − (D2
R + D2
I )
tan φ =
CRDI − CIDR
CR(2A − DR) − CIDI
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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161. Backup Slides: Radiative Seesaw with A4
MNk
extracted from diagonalized MN , and angles from diagonalization
procedure (φ, θ) related to ν mixing angles (PDG convention):
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
162. Backup Slides: Radiative Seesaw with A4
MNk
extracted from diagonalized MN , and angles from diagonalization
procedure (φ, θ) related to ν mixing angles (PDG convention):
tan2
θ12 =
cos2 θ
2
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
163. Backup Slides: Radiative Seesaw with A4
MNk
extracted from diagonalized MN , and angles from diagonalization
procedure (φ, θ) related to ν mixing angles (PDG convention):
tan2
θ12 =
cos2 θ
2
tan2
θ23 =
1/2 + cos θ sin θ cos φ
1/2 − cos θ sin θ cos φ
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
164. Backup Slides: Radiative Seesaw with A4
MNk
extracted from diagonalized MN , and angles from diagonalization
procedure (φ, θ) related to ν mixing angles (PDG convention):
tan2
θ12 =
cos2 θ
2
tan2
θ23 =
1/2 + cos θ sin θ cos φ
1/2 − cos θ sin θ cos φ
| sin θ13| =
sin θ
√
3
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
165. Backup Slides: Radiative Seesaw with A4
MNk
extracted from diagonalized MN , and angles from diagonalization
procedure (φ, θ) related to ν mixing angles (PDG convention):
tan2
θ12 =
cos2 θ
2
tan2
θ23 =
1/2 + cos θ sin θ cos φ
1/2 − cos θ sin θ cos φ
| sin θ13| =
sin θ
√
3
Dirac CP phase is: δCP = φ + α3
2 , where α3 is Majorana phase from MN3 .
E. Ma, A. Natale, and A. Rashed, Int. J. Mod. Phys. A27 (2012) 1250134, arXiv:1206.1570 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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166. Backup: µ − τ Interchange Symmetry
Consider generalized procedure to diagonalize Mν:
Md = EαUTEβMνEβUEα, where Eη = diag(eiη1 , eiη2 , eiη3 )
Only three physical phases (1 Dirac, 2 Majorana), as seen when squaring:
M2
d = E†
αU†
Mν(Mν)†
UEα
Define new mass matrix:
M2
λd =
1 0 0
0 1 0
0 0 λ
M2
d
1 0 0
0 1 0
0 0 λ
=
m2
1 0 0
0 m2
2 0
0 0 λ2m2
3
Yields:
M2
λd = E†
αIλU†
Mν(Mν)†
UIλEα
Mλd has masses from case when λ = 1, not physical mν
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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167. Backup: Scotogenic ∆(27) Models
For scotogenic models:
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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168. Backup: Scotogenic ∆(27) Models
For scotogenic models:
Minimal Content: Φ, η ∼ 11, ν ∼ 3, N ∼ 3∗, ζi ∼ 3
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
169. Backup: Scotogenic ∆(27) Models
For scotogenic models:
Minimal Content: Φ, η ∼ 11, ν ∼ 3, N ∼ 3∗, ζi ∼ 3
U(1)D Content: Φ, η1,2 ∼ 11, ν ∼ 3, NR ∼ 3, NL ∼ 3∗, ζi ∼ 3
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
170. Backup: Scotogenic ∆(27) Models
For scotogenic models:
Minimal Content: Φ, η ∼ 11, ν ∼ 3, N ∼ 3∗, ζi ∼ 3
U(1)D Content: Φ, η1,2 ∼ 11, ν ∼ 3, NR ∼ 3, NL ∼ 3∗, ζi ∼ 3
Soft-breaking terms:
fijkNiNkζ∗
k
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
171. Backup: Scotogenic ∆(27) Models
For scotogenic models:
Minimal Content: Φ, η ∼ 11, ν ∼ 3, N ∼ 3∗, ζi ∼ 3
U(1)D Content: Φ, η1,2 ∼ 11, ν ∼ 3, NR ∼ 3, NL ∼ 3∗, ζi ∼ 3
Soft-breaking terms:
fijkNiNkζ∗
k, fijk
¯NL,iNR,jζk
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
172. Backup: Scotogenic ∆(27) Models
For scotogenic models:
Minimal Content: Φ, η ∼ 11, ν ∼ 3, N ∼ 3∗, ζi ∼ 3
U(1)D Content: Φ, η1,2 ∼ 11, ν ∼ 3, NR ∼ 3, NL ∼ 3∗, ζi ∼ 3
Soft-breaking terms:
fijkNiNkζ∗
k, fijk
¯NL,iNR,jζk
Invariants (minimal example):
f1(N1N1vζ1 + N2N2vζ2 + N3N3vζ3 )
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
173. Backup: Scotogenic ∆(27) Models
For scotogenic models:
Minimal Content: Φ, η ∼ 11, ν ∼ 3, N ∼ 3∗, ζi ∼ 3
U(1)D Content: Φ, η1,2 ∼ 11, ν ∼ 3, NR ∼ 3, NL ∼ 3∗, ζi ∼ 3
Soft-breaking terms:
fijkNiNkζ∗
k, fijk
¯NL,iNR,jζk
Invariants (minimal example):
f1(N1N1vζ1 + N2N2vζ2 + N3N3vζ3 )
f2(N1N2vζ3 + N2N3vζ1 + N3N1vζ2 + N1N3vζ2 + N2N1vζ3 + N3N2vζ1 )
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
174. Backup: Scotogenic ∆(27) Models
For scotogenic models:
Minimal Content: Φ, η ∼ 11, ν ∼ 3, N ∼ 3∗, ζi ∼ 3
U(1)D Content: Φ, η1,2 ∼ 11, ν ∼ 3, NR ∼ 3, NL ∼ 3∗, ζi ∼ 3
Soft-breaking terms:
fijkNiNkζ∗
k, fijk
¯NL,iNR,jζk
Invariants (minimal example):
f1(N1N1vζ1 + N2N2vζ2 + N3N3vζ3 )
f2(N1N2vζ3 + N2N3vζ1 + N3N1vζ2 + N1N3vζ2 + N2N1vζ3 + N3N2vζ1 )
→ produces special form of Mν
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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175. Backup: Scotogenic ∆(27) Models
For Ml diagonal, re-write Mν:
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
176. Backup: Scotogenic ∆(27) Models
For Ml diagonal, re-write Mν:
−3
2 + + 3
4κ −2η+κ
2
√
2
√
3
4 κ
−2η+κ
2
√
2
3
2 + + 1
2η + 1
2κ
√
3
2
√
2
κ
√
3
4 κ
√
3
2
√
2
κ −3
2 + − η + 1
4κ
b
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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177. Backup: Scotogenic ∆(27) Models
For Ml diagonal, re-write Mν:
−3
2 + + 3
4κ −2η+κ
2
√
2
√
3
4 κ
−2η+κ
2
√
2
3
2 + + 1
2η + 1
2κ
√
3
2
√
2
κ
√
3
4 κ
√
3
2
√
2
κ −3
2 + − η + 1
4κ
b
Mν is approximately diagonal in TBM basis. To fully diagonalize utilize:
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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178. Backup: Scotogenic ∆(27) Models
For Ml diagonal, re-write Mν:
−3
2 + + 3
4κ −2η+κ
2
√
2
√
3
4 κ
−2η+κ
2
√
2
3
2 + + 1
2η + 1
2κ
√
3
2
√
2
κ
√
3
4 κ
√
3
2
√
2
κ −3
2 + − η + 1
4κ
b
Mν is approximately diagonal in TBM basis. To fully diagonalize utilize:
U =
1 θ12 θ13
−θ12 1 θ23
−θ13 −θ23 1
,
where
θ12 ≈
ζ
6
√
2
, θ13 ≈
√
3κ
2ζ
, θ23 ≈
κ
2
√
6
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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179. Backup: Scotogenic ∆(27) Models
Approximate solutions for neutrino oscillation:
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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180. Backup: Scotogenic ∆(27) Models
Approximate solutions for neutrino oscillation:
sin θ13 ≈ ±
κ
√
2ζ
, tan θ12 ≈
1
√
2
1 − ζ/6
1 + ζ/12
, tan θ23 ≈ 1 +
κ
ζ
,
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
181. Backup: Scotogenic ∆(27) Models
Approximate solutions for neutrino oscillation:
sin θ13 ≈ ±
κ
√
2ζ
, tan θ12 ≈
1
√
2
1 − ζ/6
1 + ζ/12
, tan θ23 ≈ 1 +
κ
ζ
,
∆m2
21 ≈
3
4
(8 + ζ)b2
, ∆m2
32 ≈
3
2
ζb2
,
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
182. Backup: Scotogenic ∆(27) Models
Approximate solutions for neutrino oscillation:
sin θ13 ≈ ±
κ
√
2ζ
, tan θ12 ≈
1
√
2
1 − ζ/6
1 + ζ/12
, tan θ23 ≈ 1 +
κ
ζ
,
∆m2
21 ≈
3
4
(8 + ζ)b2
, ∆m2
32 ≈
3
2
ζb2
,
mee ≈ |fa|
For central values ζ = 0.209, and κ/ζ = O(ζ)
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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183. Backup: Scotogenic ∆(27) Models
Approximate solutions for neutrino oscillation:
sin θ13 ≈ ±
κ
√
2ζ
, tan θ12 ≈
1
√
2
1 − ζ/6
1 + ζ/12
, tan θ23 ≈ 1 +
κ
ζ
,
∆m2
21 ≈
3
4
(8 + ζ)b2
, ∆m2
32 ≈
3
2
ζb2
,
mee ≈ |fa|
For central values ζ = 0.209, and κ/ζ = O(ζ)
→ potentially large corrections to approximate solutions
E. Ma, and A. Natale, Phys. Lett. B723 (2014) 403-405, arXiv:1403.6772 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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184. Backup: µ − τ Interchange Symmetry
M2
λd = E†
αU†
(1 + Ơ
)IλMνI2
λ(Mν)†
Iλ(1 + ∆)UEα,
where ∆ = U†(Iλ − I)U
(1+∆)M2
λd(1+∆)†
= OM2
physOT
→ (1 + ∆)−1
OM2
physOT
((1 + ∆)−1
)†
Numerically diagonalizing Mphys yields full PMNS matrix for λ = 1 via
UO.
Procedure: pick physical ∆mijs and fix final θ12 and θ13 to be within PDG
ranges, pick θijs for case when λ = 1, numerically determine (1 + ∆)−1O
such that Mν is diagonalized.
E. Ma, and A. Natale, and O. Popov, Phys. Lett. B746 (2015) 114-116, arXiv:1502.08023 [hep-ph].
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
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185. Backup: Scotogenic WDM
∆aµ =
k
h2
kµm2
µ
(4π)2m2
η
F2(M2
Nk
/m2
η),
where
F2(x) =
1 − 6x + 3x2 + 2x3 − 6x2 log(x)
6(1 − x)4
≈ 1/6,
as mNK
mη, ∆aµ ∝ |h|2 m2
µ
m2
η
Alexander Anderson-Natale | Radiative Neutrino Mass, Dark Matter, Flavor Symmetry, and Collider Signatures
39/39
