tev-scale leptogenesis scenarios · 12/11/2014 · sphaleron interactions. [s. yu. khlebnikov, m....
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TeV-scale Leptogenesis Scenarios
P. S. BHUPAL DEV
Consortium for Fundamental Physics, The University of Manchester
XXI DAE-BRNS High Energy Physics Symposium 2014IIT Guwahati
December 11, 2014
Outline
Motivation
TeV-scale (Resonant) Leptogenesis
Flavor Effects
Phenomenology
Conclusion
Matter-Antimatter Asymmetry
η∆B ≡ nB − nB
nγ= (6.04± 0.08)× 10−10 [Planck (2014)]
Baryogenesis
Dynamical generation of baryon asymmetry.Sakharov conditions: [A. D. Sakharov, JETP Lett. 5, 24 (1967)]
B violation.C and CP violation.Out-of-equilibrium dynamics.
SM has all these ingredients.B violation through EW sphalerons. [G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976); F. R. Klinkhamer and N. S. Manton,
Phys. Rev. D 30, 2212 (1984); V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155, 36 (1985)]
Maximal C violation due to weak interactions. CP violation in the quark sector due tothe K-M phase.Departure from thermal equilibrium at the EWPT.
However, the observed CP violation is too small (by ∼ 10 orders of magnitude).
Also, no strong first-order phase transition, unless mH . 70 GeV. [K. Kajantie, M. Laine, K.
Rummukainen and M. E. Shaposhnikov, Phys. Rev. Lett. 77, 2887 (1996); F. Csikor, Z. Fodor and J. Heitger, Phys. Rev. Lett. 82, 21 (1999)]
Requires some New Physics:Must introduce new sources of CP violation.Must either provide a departure from equilibrium (in addition to EWPT) or modify theEWPT itself.
Baryogenesis
Dynamical generation of baryon asymmetry.Sakharov conditions: [A. D. Sakharov, JETP Lett. 5, 24 (1967)]
B violation.C and CP violation.Out-of-equilibrium dynamics.
SM has all these ingredients.B violation through EW sphalerons. [G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976); F. R. Klinkhamer and N. S. Manton,
Phys. Rev. D 30, 2212 (1984); V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155, 36 (1985)]
Maximal C violation due to weak interactions. CP violation in the quark sector due tothe K-M phase.Departure from thermal equilibrium at the EWPT.
However, the observed CP violation is too small (by ∼ 10 orders of magnitude).
Also, no strong first-order phase transition, unless mH . 70 GeV. [K. Kajantie, M. Laine, K.
Rummukainen and M. E. Shaposhnikov, Phys. Rev. Lett. 77, 2887 (1996); F. Csikor, Z. Fodor and J. Heitger, Phys. Rev. Lett. 82, 21 (1999)]
Requires some New Physics:Must introduce new sources of CP violation.Must either provide a departure from equilibrium (in addition to EWPT) or modify theEWPT itself.
Baryogenesis
Dynamical generation of baryon asymmetry.Sakharov conditions: [A. D. Sakharov, JETP Lett. 5, 24 (1967)]
B violation.C and CP violation.Out-of-equilibrium dynamics.
SM has all these ingredients.B violation through EW sphalerons. [G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976); F. R. Klinkhamer and N. S. Manton,
Phys. Rev. D 30, 2212 (1984); V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155, 36 (1985)]
Maximal C violation due to weak interactions. CP violation in the quark sector due tothe K-M phase.Departure from thermal equilibrium at the EWPT.
However, the observed CP violation is too small (by ∼ 10 orders of magnitude).
Also, no strong first-order phase transition, unless mH . 70 GeV. [K. Kajantie, M. Laine, K.
Rummukainen and M. E. Shaposhnikov, Phys. Rev. Lett. 77, 2887 (1996); F. Csikor, Z. Fodor and J. Heitger, Phys. Rev. Lett. 82, 21 (1999)]
Requires some New Physics:Must introduce new sources of CP violation.Must either provide a departure from equilibrium (in addition to EWPT) or modify theEWPT itself.
Baryogenesis
Dynamical generation of baryon asymmetry.Sakharov conditions: [A. D. Sakharov, JETP Lett. 5, 24 (1967)]
B violation.C and CP violation.Out-of-equilibrium dynamics.
SM has all these ingredients.B violation through EW sphalerons. [G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976); F. R. Klinkhamer and N. S. Manton,
Phys. Rev. D 30, 2212 (1984); V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155, 36 (1985)]
Maximal C violation due to weak interactions. CP violation in the quark sector due tothe K-M phase.Departure from thermal equilibrium at the EWPT.
However, the observed CP violation is too small (by ∼ 10 orders of magnitude).
Also, no strong first-order phase transition, unless mH . 70 GeV. [K. Kajantie, M. Laine, K.
Rummukainen and M. E. Shaposhnikov, Phys. Rev. Lett. 77, 2887 (1996); F. Csikor, Z. Fodor and J. Heitger, Phys. Rev. Lett. 82, 21 (1999)]
Requires some New Physics:Must introduce new sources of CP violation.Must either provide a departure from equilibrium (in addition to EWPT) or modify theEWPT itself.
Leptogenesis
[M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986)]
Introduce SM singlet heavy Majorana neutrinos.
New source of CP violation through Yukawa couplings.
Departure from equilibrium when Yukawa interaction rates are slower than H(t).
L violation due to the Majorana nature.
L asymmetry partially converted to B asymmetry through (B + L)-violatingsphaleron interactions. [S. Yu. Khlebnikov, M. E. Shaposhnikov, Nucl. Phys. B 308, 885 (1988)]
Singlet neutrinos naturally motivated from the seesaw mechanism.
Simplest paradigm to explain the observed neutrino oscillation data.
Neutrino Oscillations
Seesaw Mechanism
LH neutrinos exactly massless in the SM to all orders in perturbation theory,because
No RH counterpart (i.e. no Dirac mass term).νL part of SU(2)L doublet. So no Majorana mass term νT
L C−1νL.Accidental B− L symmetry, even after non-perturbative effects.
Simply adding RH neutrinos leads to tiny Yukawa couplings YN . 10−12.
A more natural way is to break B− L.
Within the SM, parametrized through dimension-5 operator λij(LTi Φ)(LT
j Φ)/Λ.[S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979)]
Three tree-level realizations: Type-I, II, III seesaw mechanism.
! masses beyond the SM : tree level
Fermionic Singlet
Seesaw ( or type I)
2 x 2 = 1 + 3
! masses beyond the SM : tree level
Fermionic Triplet
Seesaw ( or type III)
2 x 2 = 1 + 3
! masses beyond the SM : tree level
2 x 2 = 1 + 3
Scalar Triplet
Seesaw ( or type II)
Fermion singlets:(type-I seesaw)
Scalar triplet:(type-II seesaw)
Fermion triplets:(type-III seesaw)
m = Y TN
1
MNYNv2 m = Y
µ
M2
v2 m = Y T
1
MYv
2
Minkowski; Gellman, Ramon, Slansky; Yanagida;Glashow; Mohapatra, Senjanovic
Magg, Wetterich; Lazarides, Shafi; Mohapatra, Senjanovic; Schechter, Valle
Foot, Lew, He, Joshi; Ma; Ma, Roy;T.H., Lin, Notari, Papucci, Strumia; Bajc, Nemevsek,
Senjanovic; Dorsner, Fileviez-Perez;....
The 3 seesaw models
L YLiLj
µHH + h.c.
L YNijNiLjH
mNi
2N c
i Ni + h.c. mi
2c
ii + h.c.
L YijiLjH
NRii (+
i ,0i ,
i )
YN Y
(++,+,0)
for example with requiresYN 1, m 0.1 eV
requires
MN 1015 GeV
with YN 106, m 0.1 eV MN TeV
µ
! masses beyond the SM : tree level
Fermionic Singlet
Seesaw ( or type I)
2 x 2 = 1 + 3
! masses beyond the SM : tree level
Fermionic Triplet
Seesaw ( or type III)
2 x 2 = 1 + 3
! masses beyond the SM : tree level
2 x 2 = 1 + 3
Scalar Triplet
Seesaw ( or type II)
Fermion singlets:(type-I seesaw)
Scalar triplet:(type-II seesaw)
Fermion triplets:(type-III seesaw)
m = Y TN
1
MNYNv2 m = Y
µ
M2
v2 m = Y T
1
MYv
2
Minkowski; Gellman, Ramon, Slansky; Yanagida;Glashow; Mohapatra, Senjanovic
Magg, Wetterich; Lazarides, Shafi; Mohapatra, Senjanovic; Schechter, Valle
Foot, Lew, He, Joshi; Ma; Ma, Roy;T.H., Lin, Notari, Papucci, Strumia; Bajc, Nemevsek,
Senjanovic; Dorsner, Fileviez-Perez;....
The 3 seesaw models
L YLiLj
µHH + h.c.
L YNijNiLjH
mNi
2N c
i Ni + h.c. mi
2c
ii + h.c.
L YijiLjH
NRii (+
i ,0i ,
i )
YN Y
(++,+,0)
for example with requiresYN 1, m 0.1 eV
requires
MN 1015 GeV
with YN 106, m 0.1 eV MN TeV
µ
! masses beyond the SM : tree level
Fermionic Singlet
Seesaw ( or type I)
2 x 2 = 1 + 3
! masses beyond the SM : tree level
Fermionic Triplet
Seesaw ( or type III)
2 x 2 = 1 + 3
! masses beyond the SM : tree level
2 x 2 = 1 + 3
Scalar Triplet
Seesaw ( or type II)
Fermion singlets:(type-I seesaw)
Scalar triplet:(type-II seesaw)
Fermion triplets:(type-III seesaw)
m = Y TN
1
MNYNv2 m = Y
µ
M2
v2 m = Y T
1
MYv
2
Minkowski; Gellman, Ramon, Slansky; Yanagida;Glashow; Mohapatra, Senjanovic
Magg, Wetterich; Lazarides, Shafi; Mohapatra, Senjanovic; Schechter, Valle
Foot, Lew, He, Joshi; Ma; Ma, Roy;T.H., Lin, Notari, Papucci, Strumia; Bajc, Nemevsek,
Senjanovic; Dorsner, Fileviez-Perez;....
The 3 seesaw models
L YLiLj
µHH + h.c.
L YNijNiLjH
mNi
2N c
i Ni + h.c. mi
2c
ii + h.c.
L YijiLjH
NRii (+
i ,0i ,
i )
YN Y
(++,+,0)
for example with requiresYN 1, m 0.1 eV
requires
MN 1015 GeV
with YN 106, m 0.1 eV MN TeV
µ
Type-I Seesaw
[Minkowski ’77; Mohapatra, Senjanovic ’79; Yanagida ’79; Gell-Mann, Ramond, Slansky ’79; Glashow ’79]
Seesaw messenger: SM singlet fermions (RH neutrinos).Have a Majorana mass term MNNT
R C−1NR, in addition to the Dirac mass MD = vYN .In the flavor basis νC
L ,NR, leads to the general structure
Mν =
(0 MD
MTD MN
)In the seesaw approximation ||ξ|| 1, where ξ ≡ MDM−1
N and ||ξ|| ≡√
Tr(ξ†ξ),Mlightν ' −MDM−1
N MTD is the light neutrino mass matrix.
ξ ≡ MDM−1N is the heavy-light neutrino mixing.
Complex YN elements can give a non-zero CP asymmetry.Leptogenesis is a cosmological consequence of the seesaw mechanism.
=⇒
Leptogenesis for Pedestrians
[W. Buchmuller, P. Di Bari and M. Plumacher, Annals Phys. 315, 305 (2005)]
Consider a hierarchical heavy neutrino spectrum (mN1 mN2 < mN3 ).
Final baryon asymmetry due to N1 decay is given by
η∆B = d · ε1 · κf
d ' 0.96× 10−2 is the dilution factor (in converting /L to /B).ε1 ≡
∑l εl1 is the CP asymmetry, where
εlα =Γ(Nα → LlΦ)− Γ(Nα → Lc
l Φc)∑
k[Γ(Nα → LkΦ) + Γ(Nα → Lc
kΦc)]
κf ≡ κ(zf ) is the final efficiency factor, where z = mN1/T and
κ(z) =
∫ z
zi
dz′D
D + SdnN1
dz′e−∫ z
z′ dz′′W(z′′)
D = ΓD/Hz (decay and inverse decay), S (scattering) and W (washout).K-factor is defined as K = ΓD(z→∞)/H(z = 1) = m1/m∗, wherem1 = (M†DMD)11/mN1 (effective neutrino mass) andm∗ = (16/3)(π5g∗/5)1/2v2/MPl ' 1.1 meV (equilibrium neutrino mass).Strong washout: K & 3 and weak washout: K 1.
Problems with Vanilla Leptogenesis
Maximal CP asymmetry is given by
εmax1 =
316π
mN1
v2
√∆m2
atm
Lower bound on mN1 : [S. Davidson and A. Ibarra, Phys. Lett. B 535, 25 (2002); W. Buchmuller, P. Di Bari and M. Plumacher, Nucl.
Phys. B 643, 367 (2002)]
mN1 > 6.4× 108 GeV(
η∆B
6× 10−10
)(0.05 eV√
∆m2atm
)κ−1
f
Experimentally inaccessible mass range!
Also leads to a lower limit on the reheat temperature Trh & 109 GeV.
In many SUSY scenarios, need Trh . 106 − 109 GeV to avoid the Gravitinoproblem. [J. R. Ellis, J. E. Kim, D. V. Nanopoulos, Phys. Lett. B 145, 181 (1984); K. Kohri, T. Moroi and A. Yotsuyanagi, Phys. Rev. D 73,
123511 (2006); V. S. Rychkov and A. Strumia, Phys. Rev. D 75, 075011 (2007)]
Problems with Vanilla Leptogenesis
Maximal CP asymmetry is given by
εmax1 =
316π
mN1
v2
√∆m2
atm
Lower bound on mN1 : [S. Davidson and A. Ibarra, Phys. Lett. B 535, 25 (2002); W. Buchmuller, P. Di Bari and M. Plumacher, Nucl.
Phys. B 643, 367 (2002)]
mN1 > 6.4× 108 GeV(
η∆B
6× 10−10
)(0.05 eV√
∆m2atm
)κ−1
f
Experimentally inaccessible mass range!
Also leads to a lower limit on the reheat temperature Trh & 109 GeV.
In many SUSY scenarios, need Trh . 106 − 109 GeV to avoid the Gravitinoproblem. [J. R. Ellis, J. E. Kim, D. V. Nanopoulos, Phys. Lett. B 145, 181 (1984); K. Kohri, T. Moroi and A. Yotsuyanagi, Phys. Rev. D 73,
123511 (2006); V. S. Rychkov and A. Strumia, Phys. Rev. D 75, 075011 (2007)]
Resonant Leptogenesis
• Resonant Leptogenesis
×NαNα
LCl
Φ†
(a)
×Nα Nβ
Φ
L LCl
Φ†
(b)
×Nα
L
Nβ
Φ†
LCl
Φ
(c)
Importance of self-energy effects (when |mN1 −mN2| ≪ mN1,2)[J. Liu, G. Segre, PRD48 (1993) 4609;
M. Flanz, E. Paschos, U. Sarkar, PLB345 (1995) 248;L. Covi, E. Roulet, F. Vissani, PLB384 (1996) 169;
...
J. R. Ellis, M. Raidal, T. Yanagida, PLB546 (2002) 228.]
Importance of the heavy-neutrino width effects: ΓNα
[A.P., PRD56 (1997) 5431; A.P. and T. Underwood, NPB692 (2004) 303.]
Warsaw, 22–27 June 2014 Flavour Covariance in Leptogenesis A. Pilaftsis
Heavy Majorana neutrino self-energy effects on the leptonic CP-asymmetry(ε-type) become dominant [M. Flanz, E. Paschos and U. Sarkar, Phys. Lett. B 345, 248 (1995); L. Covi, E. Roulet and F.
Vissani, Phys. Lett. B 384, 169 (1996)] and resonantly enhanced, even up to order 1, when∆mN ∼ ΓN mN1,2 . [A. Pilaftsis, Nucl. Phys. B 504, 61 (1997); Phys. Rev. D 56, 5431 (1997)]
The quasi-degeneracy can be naturally motivated as due to approximate breakingof some symmetry in the leptonic sector.
Heavy neutrino mass scale can be as low as the EW scale. [A. Pilaftsis and T. Underwood, Phys.
Rev. D 72, 113001 (2005)]
A potential testable scenario of leptogenesis, with implications at both Energy andIntensity Frontiers.
RL Basics
Flavor-diagonal Boltzmann equations:
nγHN
zdηNα
dz=
(1− ηN
α
ηNeq
)∑l
γNαLlΦ
nγHN
zdδηL
l
dz=∑α
(ηNα
ηNeq− 1)δγNα
LlΦ− 2
3δηL
l
∑k
[γLlΦ
LckΦ
c + γLlΦLkΦ
+ δηLk(γLkΦ
Lcl Φ
c − γLkΦLlΦ
)]where δγNα
LlΦ= εlα
∑k γ
NαLkΦ
and γNαLΦ ≡
∑l γ
NαLlΦ
=m3
Nαπ2z K1(z)ΓNα .
Total decay width is given by
ΓNα =mNα
16π
[(h† h)αα + (hc† hc)αα
].
Unstable particle mixing effects are accounted for by the resummed Yukawacouplings [A. Pilaftsis and T. Underwood, Nucl. Phys. B 692, 303 (2004); A. Pilaftsis, Phys. Rev. D 78, 013008 (2008)]
hlα = hlα − i∑β,γ
|εαβγ |hlβ
×mα(mαAαβ + mβAβα) − iRαγ [mαAγβ(mαAαγ + mγAγα) + mβAβγ(mαAγα + mγAαγ)]
m2α − m2
β+ 2im2
αAββ + 2iIm(Rαγ)[m2α|Aβγ |2 + mβmγRe(A2
βγ)]
,
where Rαβ =m2α
m2α − m2
β+ 2im2
αAββand all transition amplitudes are evaluated
on-shell with p2 = m2Nα ≡ m2
α.
RL Basics
Flavor-diagonal Boltzmann equations:
nγHN
zdηNα
dz=
(1− ηN
α
ηNeq
)∑l
γNαLlΦ
nγHN
zdδηL
l
dz=∑α
(ηNα
ηNeq− 1)δγNα
LlΦ− 2
3δηL
l
∑k
[γLlΦ
LckΦ
c + γLlΦLkΦ
+ δηLk(γLkΦ
Lcl Φ
c − γLkΦLlΦ
)]where δγNα
LlΦ= εlα
∑k γ
NαLkΦ
and γNαLΦ ≡
∑l γ
NαLlΦ
=m3
Nαπ2z K1(z)ΓNα .
Total decay width is given by
ΓNα =mNα
16π
[(h† h)αα + (hc† hc)αα
].
Unstable particle mixing effects are accounted for by the resummed Yukawacouplings [A. Pilaftsis and T. Underwood, Nucl. Phys. B 692, 303 (2004); A. Pilaftsis, Phys. Rev. D 78, 013008 (2008)]
hlα = hlα − i∑β,γ
|εαβγ |hlβ
×mα(mαAαβ + mβAβα) − iRαγ [mαAγβ(mαAαγ + mγAγα) + mβAβγ(mαAγα + mγAαγ)]
m2α − m2
β+ 2im2
αAββ + 2iIm(Rαγ)[m2α|Aβγ |2 + mβmγRe(A2
βγ)]
,
where Rαβ =m2α
m2α − m2
β+ 2im2
αAββand all transition amplitudes are evaluated
on-shell with p2 = m2Nα ≡ m2
α.
Key Diagrams
Nβ(p, s)
Φ(q)
Lk(k, r)
[hc] βk
nΦ(q)[nLr (k)]
kl Nα(p, s)
Φ(q)
Ll(k, r)
[hc]lα
Nβ(p, s)
Φc(q)
[Lc(k, r)]l
hlβ
nΦ(q)[nLr (k)]
kl Nα(p, s)
Φc(q)
[Lc(k, r)]k
h αk
Figure : Decay and Inverse Decay.
Nβ(p)
Φ(q2)
Ln(k2, r2)
Φ(q1)
Lk(k1, r1)
hnβ [hc] β
knΦ(q1)[n
Lr1(k1)]
kl
Nα(p)
Φ(q1)
Ll(k1, r1)
Φ(q2)
Lm(k2, r2)
[hc]lα h αm
Nβ(p)
Φc(q2)
[Lc(k2, r2)]m
Φ(q1)
Lk(k1, r1)
[hc] βm [hc] β
knΦ(q1)[n
Lr1(k1)]
kl
Nα(p)
Φ(q1)
Ll(k1, r1)
Φc(q2)
[Lc(k2, r2)]n
[hc]lα [hc]nα
Figure : ∆L = 0 and ∆L = 2 scattering.
RL Basics
In the narrow-width approximation, the collision rates for the ∆L = 0 and ∆L = 2scatterings are given by [F. Deppisch and A. Pilaftsis, Phys. Rev. D 83, 076007 (2011)]
γLkΦLlΦ
=∑α,β
(γNα
LΦ + γNβLΦ
)(
1− 2imNα−mNβΓNα+ΓNβ
) 2(
h∗lαhc∗kαhlβ hc
kβ + hc∗lα h∗kαhc
lβ hkβ
)[(h†h)αα + (hc† hc)αα + (h†h)ββ + (hc† hc)ββ
]2 ,
γLkΦLc
l Φc =
∑α,β
(γNα
LΦ + γNβLΦ
)(
1− 2imNα−mNβΓNα+ΓNβ
) 2(
h∗lαh∗kαhlβ hkβ + hc∗lα hc∗
kαhclβ hc
kβ
)[(h†h)αα + (hc† hc)αα + (h†h)ββ + (hc† hc)ββ
]2 .
Flavor-dependent CP-asymmetry is given by
εlα =|hlα|2 − |hc
lα|2∑k
(|hkα|2 + |hc
kα|2) =
|hlα|2 − |hclα|2
(h†h)αα + (hc† hc)αα.
In the two heavy-neutrino flavor limit, reduces to
εlα ≈Im[(h∗lαhlβ)(h†h)αβ
](h†h)αα(h†h)ββ
(m2
Nα − m2Nβ
)mNαΓ
(0)Nβ(
m2Nα − m2
Nβ
)2+(
mNαΓ(0)Nβ
)2 .
Analytic Solution
In terms of the K-factor Kα = ΓNα/(ζ(3)HN),
ηNα
ηNeq− 1 ' 1
Kαz(attractor solution).
In the kinematic regime z > zα1 ≈ 2K−1/3α ,
dδηLl
dz= z2K1(z)
(εl −
23
zKeffl δη
Ll
),
where Keffl = κl
∑α KαBlα ≡ κlKl is the effective washout parameter due to 2↔ 2
scatterings, and κl ≡∑
k
(γ
LlΦLc
kΦc +γ
LlΦLkΦ
)+γ
LlΦLc
l Φc−γ
LlΦLlΦ∑
α γNαLΦ Blα
.
In the regime z > zl2 ≈ 2(Keff
l )−1/3, the lepton asymmetry can be approximated by
δηLl (z) ' 3
2εl
Keffl z.
up to a point z = zl3 ≈ 1.25 ln(25Keff
l ), beyond which it freezes out.
Analytic Solution
independently of the initial conditions (see Fig. 1). In thisregime, the BE (4.33) becomes
d!Ll
dz¼ z2K1ðzÞ
!"l $
2
3zKl!Ll
"; (4.35)
with "l ¼P3
#¼1 "#l and Kl ¼P3
#¼1 K#B#l. We mayinclude the numerically significant RIS-subtractedcollision terms proportional to !Ll
in (4.27), by rescalingKl ! $lKl % Keff
l , where
$l %
Pk¼e;%;&
ð'Ll!LCk!y þ 'Ll!
Lk!Þ þ 'Ll!
LCl!y $ 'Ll!
Ll!
P3#¼1 '
N#
L!B#l
¼ 2X3
#;(¼1
ð "h)y#l
"h)l( þ "h)Cy
#l"h)Cl( Þ½ð "h)y "h)Þ#( þ ð "h)Cy "h)CÞ#(( þ ð "h)y
#l"h)l( $ "h)Cy
#l"h)Cl( Þ2
½ð "h) "h)yÞll þ ð "h)C "h)CyÞll(½ð "h)y "h)Þ## þ ð "h)Cy "h)CÞ## þ ð "h)y "h)Þ(( þ ð "h)Cy "h)CÞ(((
)!1$ 2i
mN#$mN(
#N#þ #N(
"$1: (4.36)
In determining the scaling factor $l, we have assumed that!Ll
* !Lk!lin (4.18), which is a valid approximation
within a given R‘L scenario under study. Note that ifonly the diagonal # ¼ ( terms representing the RIS con-tributions are considered in the sum, $l reaches its maxi-mum value, i.e. $l ¼ 1þOð"2
l Þ. We also have checkedthat in the Ll-conserving limit of the theory, the parameter$l vanishes, as it should.
As is illustrated in Fig. 1, the solution !Llto (4.35)
exhibits different behavior in the three kinematic regimes,characterized by the specific values of the parameterz ¼ mN=T:
zl2 + 2ðKeffl Þ$1=3; zl3 + 1:25 lnð25Keff
l Þ: (4.37)
For z values in the range zl2 < z < zl3, the solution !Llmay
well be approximated by
!LlðzÞ ¼ 3
2
"l
Keffl z
: (4.38)
For z > zl3, the lepton-number density !Llfreezes out and
approaches the constant value !Ll¼ ð3"lÞ=ð2Keff
l zl3Þ.2The general behavior of !Ll
in the different regimes isdisplayed in Fig. 1.In this paper we only consider R‘L scenarios, for which
the washout is strong enough, such that the critical tempera-ture zc ¼ mN=Tc where the baryon asymmetry !B decou-ples from the lepton asymmetries!Ll
is situated in the lineardropoff or constant regime. Specifically, we require that
zc > 2K$1=3# ; zc > 2ðKeff
l Þ$1=3; (4.39)
for all heavy-neutrino species N# ¼ N1;2;3 and lepton fla-vors l ¼ e, %, &. As a consequence, the baryon asymmetry!B becomes relatively independent of the initial values of!Ll
and !N#. In this case, taking into account all factors in
(4.29), (4.30), and (4.38), the resulting BAU is estimatedto be
!B ¼ $ 28
51
1
27
3
2
X
l¼e;%;&
"l
Keffl minðzc; zl3Þ
+ $3 , 10$2X
l¼e;%;&
"l
Keffl min½mN=Tc; 1:25 lnð25Keff
l Þ( :
(4.40)
We note that the formula (4.40) provides a fairly goodestimate of the BAU !B to less than 20%, in the applicableregime of approximations given by (4.39) forKeff
l * 5 for aright-handed neutrino mass scale of the order of the electro-weak scale. Hence, to account for the observed BAU !obs
Bgiven in (4.31), lepton asymmetries "l * 10$7 are required.In the next section, we present numerical estimates of the
z1 z2 z3zc
N1
L
10 2 10 1 100 101 102
10 10
10 9
10 8
10 7
10 6
10 5
10 4
z
L,
N
FIG. 1 (color online). Numerical solutions to BEs (4.32) and(4.35) for "!N1
¼ !N1=!eq
N $ 1 and !L&, respectively, in a R&L
model with mN ¼ 220 GeV, K1 ¼ 106, Keff& ¼ 102, "& ¼ 10$6
and zc ¼ mN=Tc ¼ 1:6.
2The onset of the freeze-out is defined as the position zl3 wherethe relative slope of (4.35) drops below 1, i.e. when j!0
Ll=!Ll
j ¼2=3ðzl3Þ3K1ðzl3ÞKeff
l ¼ 1. The solution to this equation can beanalytically expressed in terms of the Lambert W function, towhich zl3 in (4.37) proves to be an excellent interpolatingapproximation over the wide range of values Keff
l + 2$ 1010.
FRANK F. DEPPISCH AND APOSTOLOS PILAFTSIS PHYSICAL REVIEW D 83, 076007 (2011)
076007-12
[F. Deppisch and A. Pilaftsis, Phys. Rev. D 83, 076007 (2011)]
Flavordynamics of RL
Flavor effects important for the time-evolution of lepton asymmetry in RL models.Two sources of flavor effects, due to
Heavy neutrino Yukawa couplings h αl .[A. Pilaftsis, PRL 95, 081602 (2005); T. Endoh, T. Morozumi and Z.-h. Xiong, PTP 111, 123 (2004); P. Di Bari, NPB 727, 318 (2005);
S. Blanchet, P. Di Bari, D. A. Jones and L. Marzola, JCAP 1301, 041 (2013)]
Charged lepton Yukawa couplings y kl . [R. Barbieri, P. Creminelli, A. Strumia and N. Tetradis, NPB 575, 61 (2000); A.
Abada, S. Davidson, F. -X. Josse-Michaux, M. Losada and A. Riotto, JCAP 0604, 004 (2006); E. Nardi, Y. Nir, E. Roulet and J. Racker, JHEP
0601, 164 (2006); S. Blanchet and P. Di Bari, JCAP 0703, 018 (2007)]
Lead to three distinct physical phenomena: mixing, oscillation and (de)coherence.
Fully flavor-covariant formalism essential to capture consistently all flavor effects.
Within the semi-classical Boltzmann framework, we promote individual numberdensities to number density matrices (the so-called ’density matrix’ formalism).[G. Sigl and G. Raffelt, Nucl. Phys. B 406, 423 (1993)]
Obtain manifestly flavor-covariant transport equations.[PSBD, P. Millington, A. Pilaftsis and D. Teresi, Nucl. Phys. B 886, 569 (2014)]
Flavor Covariant Formalism
Unitary flavor transformations:
Ll → V ml Lm, L†,l → V l
m L†,m, NR,α → U βα NR,β , N†,αR → Uα
β N†,βR .
The Lagrangian
−LN = h αl LlΦ NR,α + NC
R,α [MN ]αβ NR,β + H.c.
is invariant if h αl → V ml Uα
β h βm , [MN ]αβ → Uα
γ Uβδ [MN ]γδ.
Flavor-covariant quantization, e.g.
Ll(x) =
∫p,s
[(2EL(p))−
12
] i
l
([e−ip·x] j
i[u(p, s)] k
j bk(p, s) +[eip·x] j
i[v(p, s)] k
j d†k (p, s))
Matrix number densities: [nL] ml ∝ 〈b†,m bl〉, [nL] m
l ∝ 〈d†l dm〉, [nN ] βα ∝ 〈a†,β aα〉
Necessary to consider generalized discrete symmetries C,P, T, e.g.
dl = (bl)C ≡ G†,lm(bl)
C with G = V VT.
Number densities transform as (nL)C = (nL)T, (nN)C = (nN)T(nN , nN not independent)
Define CP-“even” and CP-“odd” quantities:
nN =12
(nN + nN), δnN = nN − nN , δnL = nL − nL.
Flavor Covariant Formalism
Unitary flavor transformations:
Ll → V ml Lm, L†,l → V l
m L†,m, NR,α → U βα NR,β , N†,αR → Uα
β N†,βR .
The Lagrangian
−LN = h αl LlΦ NR,α + NC
R,α [MN ]αβ NR,β + H.c.
is invariant if h αl → V ml Uα
β h βm , [MN ]αβ → Uα
γ Uβδ [MN ]γδ.
Flavor-covariant quantization, e.g.
Ll(x) =
∫p,s
[(2EL(p))−
12
] i
l
([e−ip·x] j
i[u(p, s)] k
j bk(p, s) +[eip·x] j
i[v(p, s)] k
j d†k (p, s))
Matrix number densities: [nL] ml ∝ 〈b†,m bl〉, [nL] m
l ∝ 〈d†l dm〉, [nN ] βα ∝ 〈a†,β aα〉
Necessary to consider generalized discrete symmetries C,P, T, e.g.
dl = (bl)C ≡ G†,lm(bl)
C with G = V VT.
Number densities transform as (nL)C = (nL)T, (nN)C = (nN)T(nN , nN not independent)
Define CP-“even” and CP-“odd” quantities:
nN =12
(nN + nN), δnN = nN − nN , δnL = nL − nL.
Flavor Covariant Formalism
Unitary flavor transformations:
Ll → V ml Lm, L†,l → V l
m L†,m, NR,α → U βα NR,β , N†,αR → Uα
β N†,βR .
The Lagrangian
−LN = h αl LlΦ NR,α + NC
R,α [MN ]αβ NR,β + H.c.
is invariant if h αl → V ml Uα
β h βm , [MN ]αβ → Uα
γ Uβδ [MN ]γδ.
Flavor-covariant quantization, e.g.
Ll(x) =
∫p,s
[(2EL(p))−
12
] i
l
([e−ip·x] j
i[u(p, s)] k
j bk(p, s) +[eip·x] j
i[v(p, s)] k
j d†k (p, s))
Matrix number densities: [nL] ml ∝ 〈b†,m bl〉, [nL] m
l ∝ 〈d†l dm〉, [nN ] βα ∝ 〈a†,β aα〉
Necessary to consider generalized discrete symmetries C,P, T, e.g.
dl = (bl)C ≡ G†,lm(bl)
C with G = V VT.
Number densities transform as (nL)C = (nL)T, (nN)C = (nN)T(nN , nN not independent)
Define CP-“even” and CP-“odd” quantities:
nN =12
(nN + nN), δnN = nN − nN , δnL = nL − nL.
Flavor Covariant Transport Equations
nX(t) ≡ 〈nX (t; ti)〉t = Trρ(t; ti) nX (t; ti)
.
Markovian master equation for number density matrices:
ddt
nX(k, t) ' i〈 [HX0 , nX(k, t)] 〉t −
12
∫ +∞
−∞dt′ 〈 [Hint(t′), [Hint(t), nX(k, t)]] 〉t.
For charged-lepton and heavy-neutrino matrix number densities, we find:
ddt
[nLs1s2 (p, t)] m
l = − i[EL(p), nL
s1s2 (p, t)] m
l+ [CL
s1s2 (p, t)] ml
ddt
[nNr1r2 (k, t)] β
α = − i[EN(k), nN
r1r2 (k, t)] β
α+ [CN
r1r2 (k, t)] βα + Gαλ [C
Nr2r1 (k, t)] λ
µ Gµβ
Collision terms are of the form
[CLs1s2 (p, t)] m
l ⊃ −12
[Fs1s r1r2 (p, q, k, t)] n βl α [Γs s2r2r1 (p, q, k)] m α
n β ,
where F = nΦ nL ⊗(1− nN) − (
1 + nΦ) (
1− nL)⊗ nN are the statistical tensors, andΓ are the rank-4 absorptive rate tensors describing heavy neutrino decays and inversedecays.
Collision Rates for Decay and Inverse Decay
nΦ [nL] kl [γ(LΦ → N)] l β
k α
L
Φ
Nβ Nα
[hc] βk
[hc]lα
↓
Nβ(p, s)
Φ(q)
Lk(k, r)
[hc] βk
nΦ(q)[nLr (k)]
kl Nα(p, s)
Φ(q)
Ll(k, r)
[hc]lα
Collision Rates for 2 ↔ 2 Scattering
nΦ [nL] kl [γ(LΦ → LΦ)] l n
k m
Φ
L
ΦLn Lm
hnβ h α
m
[hc] βk [hc]lα
Nβ Nα
↓
Nβ(p)
Φ(q2)
Ln(k2, r2)
Φ(q1)
Lk(k1, r1)
hnβ [hc] β
knΦ(q1)[n
Lr1(k1)]
kl
Nα(p)
Φ(q1)
Ll(k1, r1)
Φ(q2)
Lm(k2, r2)
[hc]lα h αm
Application to Resonant Leptogenesis
Classical statistics: 1± nX ' 1.
Kinetic equilibrium.
Degenerate spin degrees of freedom.
Small deviation from equilibrium for charged leptons: [nL] ml + [nL] m
l ' 2 nLeq δ
ml .
Unstable particle mixing accounted for by resummed Yukawa couplings:h αl → h αl . [A. Pilaftsis and T. Underwood, Nucl. Phys. B 692, 303 (2004)]
[γNLΦ]
m β
l α ∝ hmαh βl + [hc]m
α[hc] βl
[δγNLΦ]
m β
l α ∝ hmαh βl − [hc]m
α[hc] βl .
Thermal RIS-subtraction to avoid double counting.[E. W. Kolb and S. Wolfram, Nucl. Phys. B 172, 224 (1980)]
Charged-lepton decoherence processes involving the RH charged leptons (otherscan be ignored by virtue of the KMS relation).
Final Rate Equations
HN nγ
zd[ηN ] β
α
dz= − i
nγ
2
[EN , δη
N] β
α+[Re(γN
LΦ)] β
α− 1
2 ηNeq
ηN , Re(γN
LΦ) β
α
HN nγ
zd[δηN ] β
α
dz= − 2 i nγ
[EN , η
N] β
α+ 2 i
[Im(δγN
LΦ)] β
α− i
ηNeq
ηN , Im(δγN
LΦ) β
α
− 12 ηN
eq
δηN , Re(γN
LΦ) β
α
HN nγ
zd[δηL] m
l
dz= − [δγN
LΦ]m
l +[ηN ] α
β
ηNeq
[δγNLΦ]
m β
l α +[δηN ] α
β
2 ηNeq
[γNLΦ]
m β
l α
− 13
δηL, γLΦ
LcΦc + γLΦLΦ
m
l− 2
3[δηL]
nk
([γLΦ
LcΦc ]k m
n l − [γLΦLΦ ]
k mn l
)− 2
3
δηL, γdec
m
l+ [δγback
dec ] ml
Final Rate Equations: Mixing
HN nγ
zd[ηN ] β
α
dz= − i
nγ
2
[EN , δη
N] β
α+[Re(γN
LΦ)] β
α− 1
2 ηNeq
ηN , Re(γN
LΦ) β
α
HN nγ
zd[δηN ] β
α
dz= − 2 i nγ
[EN , η
N] β
α+ 2 i
[Im(δγN
LΦ)] β
α− i
ηNeq
ηN , Im(δγN
LΦ) β
α
− 12 ηN
eq
δηN , Re(γN
LΦ) β
α
HN nγ
zd[δηL] m
l
dz= − [δγN
LΦ]m
l +[ηN ] α
β
ηNeq
[δγNLΦ]
m β
l α +[δηN ] α
β
2 ηNeq
[γNLΦ]
m β
l α
− 13
δηL, γLΦ
LcΦc + γLΦLΦ
m
l− 2
3[δηL]
nk
([γLΦ
LcΦc ]k m
n l − [γLΦLΦ ]
k mn l
)− 2
3
δηL, γdec
m
l+ [δγback
dec ] ml
Final Rate Equations: Oscillation
HN nγ
zd[ηN ] β
α
dz= − i
nγ
2
[EN , δη
N] β
α+[Re(γN
LΦ)] β
α− 1
2 ηNeq
ηN , Re(γN
LΦ) β
α
HN nγ
zd[δηN ] β
α
dz= − 2 i nγ
[EN , η
N] β
α+ 2 i
[Im(δγN
LΦ)] β
α− i
ηNeq
ηN , Im(δγN
LΦ) β
α
− 12 ηN
eq
δηN , Re(γN
LΦ) β
α
HN nγ
zd[δηL] m
l
dz= − [δγN
LΦ]m
l +[ηN ] α
β
ηNeq
[δγNLΦ]
m β
l α +[δηN ] α
β
2 ηNeq
[γNLΦ]
m β
l α
− 13
δηL, γLΦ
LcΦc + γLΦLΦ
m
l− 2
3[δηL]
nk
([γLΦ
LcΦc ]k m
n l − [γLΦLΦ ]
k mn l
)− 2
3
δηL, γdec
m
l+ [δγback
dec ] ml
Notice:
O(h4) in total lepton asymmetry.Hence, different from the ARSmechanism, which is an O(h6) effect.[E. Akhmedov, V. Rubakov, A. Smirnov, PRL 81, 1359 (1998);T. Asaka and M. Shaposhnikov, PLB 620, 17 (2005);B. Shuve and I. Yavin, arXiv:1401.2459 [hep-ph]]
Final Rate Equations: Oscillation
HN nγ
zd[ηN ] β
α
dz= − i
nγ
2
[EN , δη
N] β
α+[Re(γN
LΦ)] β
α− 1
2 ηNeq
ηN , Re(γN
LΦ) β
α
HN nγ
zd[δηN ] β
α
dz= − 2 i nγ
[EN , η
N] β
α+ 2 i
[Im(δγN
LΦ)] β
α− i
ηNeq
ηN , Im(δγN
LΦ) β
α
− 12 ηN
eq
δηN , Re(γN
LΦ) β
α
HN nγ
zd[δηL] m
l
dz= − [δγN
LΦ]m
l +[ηN ] α
β
ηNeq
[δγNLΦ]
m β
l α +[δηN ] α
β
2 ηNeq
[γNLΦ]
m β
l α
− 13
δηL, γLΦ
LcΦc + γLΦLΦ
m
l− 2
3[δηL]
nk
([γLΦ
LcΦc ]k m
n l − [γLΦLΦ ]
k mn l
)− 2
3
δηL, γdec
m
l+ [δγback
dec ] ml
Notice:
O(h4) in total lepton asymmetry.Hence, different from the ARSmechanism, which is an O(h6) effect.[E. Akhmedov, V. Rubakov, A. Smirnov, PRL 81, 1359 (1998);T. Asaka and M. Shaposhnikov, PLB 620, 17 (2005);B. Shuve and I. Yavin, arXiv:1401.2459 [hep-ph]]
Final Rate Equations: Charged Lepton Decoherence
HN nγ
zd[ηN ] β
α
dz= − i
nγ
2
[EN , δη
N] β
α+[Re(γN
LΦ)] β
α− 1
2 ηNeq
ηN , Re(γN
LΦ) β
α
HN nγ
zd[δηN ] β
α
dz= − 2 i nγ
[EN , η
N] β
α+ 2 i
[Im(δγN
LΦ)] β
α− i
ηNeq
ηN , Im(δγN
LΦ) β
α
− 12 ηN
eq
δηN , Re(γN
LΦ) β
α
HN nγ
zd[δηL] m
l
dz= − [δγN
LΦ]m
l +[ηN ] α
β
ηNeq
[δγNLΦ]
m β
l α +[δηN ] α
β
2 ηNeq
[γNLΦ]
m β
l α
− 13
δηL, γLΦ
LcΦc + γLΦLΦ
m
l− 2
3[δηL]
nk
([γLΦ
LcΦc ]k m
n l − [γLΦLΦ ]
k mn l
)− 2
3
δηL, γdec
m
l+ [δγback
dec ] ml
Captures three distinct physical phenomena: mixing, oscillation and decoherence.
Final Rate Equations: Charged Lepton Decoherence
HN nγ
zd[ηN ] β
α
dz= − i
nγ
2
[EN , δη
N] β
α+[Re(γN
LΦ)] β
α− 1
2 ηNeq
ηN , Re(γN
LΦ) β
α
HN nγ
zd[δηN ] β
α
dz= − 2 i nγ
[EN , η
N] β
α+ 2 i
[Im(δγN
LΦ)] β
α− i
ηNeq
ηN , Im(δγN
LΦ) β
α
− 12 ηN
eq
δηN , Re(γN
LΦ) β
α
HN nγ
zd[δηL] m
l
dz= − [δγN
LΦ]m
l +[ηN ] α
β
ηNeq
[δγNLΦ]
m β
l α +[δηN ] α
β
2 ηNeq
[γNLΦ]
m β
l α
− 13
δηL, γLΦ
LcΦc + γLΦLΦ
m
l− 2
3[δηL]
nk
([γLΦ
LcΦc ]k m
n l − [γLΦLΦ ]
k mn l
)− 2
3
δηL, γdec
m
l+ [δγback
dec ] ml
Captures three distinct physical phenomena: mixing, oscillation and decoherence.
RL from ‘First Principles’
Use Kadanoff-Baym equations for an exact description of quantum transportphenomena.To define number densities, often have to perform a gradient expansion intime-derivatives.Implicitly discards absorptive transitions.We perform a perturbative loopwise truncation of the KB equations using aperturbative approach to non-equilibrium QFT. [P. Millington and A. Pilaftsis, Phys. Rev. D 88, 085009 (2014)]
Confirm that mixing and oscillation are distinct phenomena, and both are ofsimilar magnitude in RL. [PSBD, P. Millington, A. Pilaftsis and D. Teresi, Nucl. Phys. B, to appear (2014)]
Field-Theoretic Transport PhenomenaMixing and Oscillations
0.2 0.5 110-8
10-7
10-6
z = mNêT
dhL
dhL
dhmixL
dhoscL
Combination ”÷L = ”÷Losc + ”÷L
mix yields a factor of 2 enhancementcompared to the isolated contributions for weakly-resonant RL.
Testing TeV-scale Leptogenesis at Energy Frontier
Through detection of a heavy neutrino (Majorana/(pseudo)-Dirac).q
q′
W+
ℓ+
Nℓ+
W− j
j
q
q′
W+
l+
Nl−
W+l+
ν
There are additional t-channel infra-red enhanced contributions which areimportant for LHC. [PSBD, A. Pilaftsis, U. K. Yang, Phys. Rev. Lett. 112, 081801 (2014)]
q
q′W+
γ
ℓ+
ℓ+
N
j
j
q
q′
W+
γ
j
j
W−
ℓ+
N
q
q′q′
γ
j
j
W+
ℓ+
N
q
q′
q′′
γ
j
j
W+ℓ+
N
Discovery of LNV at the LHC would rule out high-scale leptogenesis.[F. Deppisch, J. Harz and M. Hirsch, Phys. Rev. Lett. 112, 221601 (2014)]
Testing TeV-scale Leptogenesis at Intensity Frontier
Observable BP1 BP2 BP3 Exp. Limit
BR(µ→ eγ) 4.5× 10−15 1.9× 10−13 2.3× 10−17 < 5.7× 10−13
BR(τ → µγ) 1.2× 10−17 1.6× 10−18 8.1× 10−22 < 4.4× 10−8
BR(τ → eγ) 4.6× 10−18 5.9× 10−19 3.1× 10−22 < 3.3× 10−8
BR(µ→ 3e) 1.5× 10−16 9.3× 10−15 4.9× 10−18 < 1.0× 10−12
RTiµ→e 2.4× 10−14 2.9× 10−13 2.3× 10−20 < 6.1× 10−13
RAuµ→e 3.1× 10−14 3.2× 10−13 5.0× 10−18 < 7.0× 10−13
RPbµ→e 2.3× 10−14 2.2× 10−13 4.3× 10−18 < 4.6× 10−11
|Ω|eµ 5.8× 10−6 1.8× 10−5 1.6× 10−7 < 7.0× 10−5
〈m〉 [eV] 3.8× 10−3 3.8× 10−3 3.8× 10−3 < (0.11–0.25)
[PSBD, P. Millington, A. Pilaftsis and D. Teresi, Nucl. Phys. B 886, 569 (2014)]
TeV-scale Leptogenesis in Left-Right Seesaw
Common lore: MWR > 18 TeV for successful leptogenesis. [J.-M. Frere, T. Hambye and G. Vertongen,
JHEP 0901, 051 (2009)]
True only in the Minimal LRSM with generic YN . 10−5.5.
In the class of TeV-scale LRSM with large MD elements [PSBD, C.-H. Lee and R. N. Mohapatra, Phys.
Rev. D 88, 093010 (2013)], including flavor effects substantially lowers the leptogenesis boundto MWR & 3 TeV. [PSBD, C.-H. Lee and R. N. Mohapatra, Phys. Rev. D 90, 095012 (2014)]
10-4 10-3 10-2 0.1 110-11
10-10
10-9
10-8
10-7
Y
|ηΔL(zc)|
MWR= 3 TeV
MWR= 4 TeV
MWR= 6 TeV
MWR= 8 TeV
MWR= 10 TeV
|ηobsΔL | = 2.47× 10-8
2 4 6 8 1010-12
10-11
10-10
10-9
10-8
10-7
10-6
MWR(TeV)
|ηΔL(zc)|
δM = 15 MeV
δM = 1 MeV
δM = 100MeV
δM = 1 GeV
δM = 10 GeV
|ηobsΔL | = 2.47× 10-8
Possible correlations of the lepton asymmetry with LFV is an interesting probe ofTeV-scale leptogenesis in these models.
Conclusion
Leptogenesis provides an attractive link between two seemingly disparate piecesof evidence for BSM physics, namely, neutrino mass and baryon asymmetry.
Resonant Leptogenesis provides a way to test this idea in laboratory experiments.
Flavor effects play a crucial role in obtaining a realiable estimate of the leptonasymmetry.
A fully flavor-covariant formalism essential to describe consistently all flavoreffects.
Provides a complete and unified description of Resonant Leptogenesis, capturingthree physically distinct phenomena:
resonant mixing between heavy neutrinos,coherent oscillations between heavy-neutrino flavors,quantum decoherence effects in the charged-lepton sector.
Mixing and oscillation are both important and should be taken into account in RLcalculations.
TeV-scale leptogenesis leads to testable effects at both Energy and Intensityfrontiers.
Backup slides
Minimal Model of Resonant τ -Genesis
Consider a minimal model of Resonant τ -Genesis. [A. Pilaftsis, Phys. Rev. Lett. 95, 081602 (2005)]
Yukawa couplings break the O(3)-symmetry to (almost) U(1)Le+Lµ × U(1)Lτ :
h =
0 ae−iπ/4 aeiπ/4
0 be−iπ/4 beiπ/4
0 0 0
+δh, where δh =
εe 0 0εµ 0 0
ετ κ1e−i(π4 −γ1) κ2ei(π4 −γ2)
From type-I seesaw formula: Mν ' − v2
2 h M−1N hT = 0 in the limit δh→ 0.
To satisfy neutrino oscillation data, require
a2 =2mN
v2κN
(Mν,11 −
M2ν,13
Mν,33
), b2 =
2mN
v2κN
(Mν,22 −
M2ν,23
Mν,33
),
ε2e =
2mN
v2
M2ν,13
Mν,33, ε2
µ =2mN
v2
M2ν,23
Mν,33, ε2
τ =2mN
v2 Mν,33 .
where κN ≡ 18π2 ln
(µXmN
) [2κ1κ2 sin(γ1 + γ2) + i(κ2
2 − κ21)].
Input model parameters: mN , κ1, κ2, γ1, γ2.
Benchmark Points
Parameters BP1 BP2 BP3
mN 120 GeV 400 GeV 5 TeVγ1 π/4 π/3 3π/8γ2 0 0 π/2κ1 4× 10−5 2.4× 10−5 2× 10−4
κ2 2× 10−4 6× 10−5 2× 10−5
a (7.41− 5.54 i)× 10−4 (4.93− 2.32 i)× 10−3 (4.67 + 4.33 i)× 10−3
b (1.19− 0.89 i)× 10−3 (8.04− 3.79 i)× 10−3 (7.53 + 6.97 i)× 10−3
εe 3.31× 10−8 5.73× 10−8 2.14× 10−7
εµ 2.33× 10−7 4.3× 10−7 1.5× 10−6
ετ 3.5× 10−7 6.39× 10−7 2.26× 10−6
Observable BP1 BP2 BP3 Exp. Limit
BR(µ→ eγ) 4.5× 10−15 1.9× 10−13 2.3× 10−17 < 5.7× 10−13
BR(τ → µγ) 1.2× 10−17 1.6× 10−18 8.1× 10−22 < 4.4× 10−8
BR(τ → eγ) 4.6× 10−18 5.9× 10−19 3.1× 10−22 < 3.3× 10−8
BR(µ→ 3e) 1.5× 10−16 9.3× 10−15 4.9× 10−18 < 1.0× 10−12
RTiµ→e 2.4× 10−14 2.9× 10−13 2.3× 10−20 < 6.1× 10−13
RAuµ→e 3.1× 10−14 3.2× 10−13 5.0× 10−18 < 7.0× 10−13
RPbµ→e 2.3× 10−14 2.2× 10−13 4.3× 10−18 < 4.6× 10−11
|Ω|eµ 5.8× 10−6 1.8× 10−5 1.6× 10−7 < 7.0× 10−5
〈m〉 [eV] 3.8× 10−3 3.8× 10−3 3.8× 10−3 < (0.11–0.25)
Lepton Asymmetry for BP 2
-∆ΗL
+∆ΗL
-∆ΗobsL
mN = 400 GeV
zc0.2 1 10 2010-9
10-8
10-7
10-6
10-5
z = mNT
±∆Η
L
total, ΗinN=0, ∆Ηin
L=I
total, ΗinN=0, ∆Ηin
L=0
total, ΗinN=Ηeq
N , ∆ΗinL=0
N diag., ΗinN=Ηeq
N , ∆ΗinL=0
N diag., analytic
L diag., ΗinN=Ηeq
N , ∆ΗinL=0
N, L diag., ΗinN=Ηeq
N , ∆ΗinL=0
BP 2: Lepton Flavor Content
-dhobsL
mN = 400 GeV
zc0.2 1 10 2010-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
z = mNT
¨dhlmL
¨ -dhtt
L
+dhmm
L
-dheeL
¨dhtm
L ¨¨dh
teL ¨
¨dhmeL ¨
BP 2: Heavy Neutrino Flavor Content
mN = 400 GeV
zc0.2 1 10 2010-14
10-12
10-10
10-8
10-6
10-4
10-2
100
¨h abNh eqN
-d
ab¨
h11N heq
N - 1
h22N heq
N - 1
h33N heq
N - 1
¨h12N heq
N ¨¨h13
N heqN ¨
¨h23N heq
N ¨
BP 1
-∆ΗL
+∆ΗL
-∆ΗobsL
mN = 120 GeV
zc0.2 1 10 2010-9
10-8
10-7
10-6
10-5
10-4
z = mNT
±∆Η
L
total, ΗinN=0, ∆Ηin
L=I
total, ΗinN=0, ∆Ηin
L=0
total, ΗinN=Ηeq
N , ∆ΗinL=0
N diag., ΗinN=Ηeq
N , ∆ΗinL=0
N diag., analytic
L diag., ΗinN=Ηeq
N , ∆ΗinL=0
N, L diag., ΗinN=Ηeq
N , ∆ΗinL=0
BP 3
-∆ΗL
+∆ΗL
-∆ΗobsL
0.2 1 10 2010-8
10-7
10-6
10-5
10-4
10-3
z = mNT
±∆Η
L
mN = 5 TeV total, ΗinN=0, ∆Ηin
L=I
total, ΗinN=0, ∆Ηin
L=0
total, ΗinN=Ηeq
N , ∆ΗinL=0
N diag., ΗinN=Ηeq
N , ∆ΗinL=0
N diag., analytic
L diag., ΗinN=Ηeq
N , ∆ΗinL=0
N, L diag., ΗinN=Ηeq
N , ∆ΗinL=0