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bla
Conformal Standard Model
Hermann Nicolai
MPI fur Gravitationsphysik, Potsdam
(Albert Einstein Institut)
Based on: K. Meissner and HN, PLB648(2007)090001
and further joint work (in various combinations) with
A. Latosinski, A. Lewandowski, K. Meissner and P. Chankowski:
Mod.Phys.Lett.A30(2015)1550006;
JHEP1510(2015)170; Phys.Rev. D79(2018)035024
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Basic issues and philosophy
No signs of ‘new physics’ at LHC: which is the correct
road to take SM consistently up to Planck scale?
SM already fairly complete except for:
• Hierarchy problem?
• Instability/metastability of electroweak vacuum?
• Matter/antimatter asymmetry (leptogenesis)?
• Dark Matter?
‘Grand problems’ left for Quantum Gravity to solve:
• UV completion of SM physics?
• Cosmological Constant and Dark Energy?
• Origin of Inflation?
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A basic question (Ockham’s razor)
What is the minimal extension required tokeep the SM viable up to MPL?
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Related/previous work (not complete...)
• Survival of SM up to MPL [Froggat,Nielsen(1996),Jegerlehner,...]
• νMSM model [Shaposhnikov;Asaka,Bezrukov,Blanchet,Gorbunov,Shaposhnikov;...]
• Coleman-Weinberg symmetry breaking[Elias et al.(2003); Hempfling(1996);Foot et al.(2010);...]
• Conformal symmetry and electroweak hierarchy[Bardeen (1995);Meissner,HN(2007);Holthausen, Kubo, Lindner, Smirnov;...]
• Conformal models with (B − L) gauging[Iso,Okada,Orikasa(2009); + Takahashi(2015)]
• Asymptotic safety and conformal fixed point at MPL
[Wetterich,Shaposhnikov(2010);....]
• ......
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The electroweak hierarchy problem
A main focus of BSM model building over many years:
m2R = m2
B + δm2B , δm2
B ∝ Λ2 and m2R ≪ Λ2
But is this really a problem?
• Not in renormalized perturbation theory becauseΛ → ∞ and because renormalisation “does not care”whether an infinity is quadratic or logarithmic!(as exemplified by dimensional regularisation which doesnot even “see” quadratic divergences for d = 4 + ε).
• Yes, if SM is embedded into Planck scale theoryand Λ is a physical scale (cutoff) ⇒ quadratic de-pendencies on cutoff imply extreme sensitivity oflow energy physics to Planck scale physics.
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Possible solutions?
• Low energy supersymmetry: exact cancellation ofquadratic divergences ensured by (softly broken)supersymmetry ⇒ choice of cutoff Λ does not mat-ter, can formally send Λ → ∞ and adopt any conve-nient renormalisation scheme.
• Technicolor (motivated by QCD): no fundamentalscalars ⇒ no quadratic divergences ⇒ Higgs bosonwould have to be composite (could still be true...)
• Asymptotic Safety: UV problems are cured by non-trivial (non-Gaussian) fixed point of RG equations.
NB: if true, QFT remains valid at all scales, no fancy schemes
(like string theory) required for UV completion of SM physics!
• Some variant of Conformal Symmetry → This Talk!
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Large scales other than MPL ?
• (SUSY?) Grand Unification:
– Unification of gauge couplings at mX ≥ O(1016GeV)?
– But: proton refuses to decay (so far, at least!)
• Light neutrinos (mν ≤ O(1 eV)) and heavy neutrinos→ most popular (and most plausible) explanationof observed mass patterns via seesaw mechanism:[Minkowski(1977);Gell-Mann,Ramond,Slansky(1979);Yanagida(1980)]
m(1)ν ∼ m2
D/M mD = O(mW ) ⇒ m(2)ν ∼ M ≥ O(1012GeV)?
• Strong CP problem ⇒ need axion a(x)?Limits e.g. from axion cooling in stars ⇒
L =1
4faaF µνFµν with fa ≥ O(1010GeV)
→ axion is (still) an attractive CDM candidate.
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Low energy supersymmetry?
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Low energy exotics?
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Absence of any evidence from LHC for ‘new physics’raises doubts about many suggested BSM scenarios ⇒
Can the SM survive all the way to Planck scale MPL?
In this talk: explore (softly broken) conformal sym-metry for a minimal extension of usual SM as an al-ternative option.
NB: this proposal does without low energy supersym-metry, but supersymmetry may still be essential for afinite and consistent theory of quantum gravity (thusmoving the SUSY scale back up to the Planck scale).
Important: no assumptions about Planck scale theory
(assumed to ‘take over’ from QFT at MPL ∼ 1019GeV)
other than its UV finiteness (≡ UV completeness).
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Reminder: the conformal group SO(2, 4)
This is an old subject! [for a review, see H.Kastrup, arXiv:0808.2730]
Conformal group = extension of Poincare group (withgenerators Mµν, Pµ) by five more generators D and Kµ:
• Dilatations (D) : xµ → eωxµ
• Special conformal transformations (Kµ):
x′µ=
xµ − x2 · cµ1− 2c · x + c2x2
eiωDP µPµe−iωD = e2ωP µPµ ⇒ exact conformal invariance
implies that one-particle spectrum is either continuous(= R+) or consists only of the single point {0}.
Consequently, conformal group cannot be realized asan exact symmetry in nature.
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Conformal Invariance and the Standard Model
Fact: Standard Model of elementary particle physicsis conformally invariant at tree level except for explicitmass term m2Φ†Φ in potential ⇒Masses for vector bosons, quarks and leptons →Can ‘softly broken conformal symmetry’ (≡ ‘SBCS’)stabilize the electroweak scale w.r.t. the Planck scale?
Concrete implementation of this idea requires
• Absence of any intermediate mass scales betweenMEW and MPL (‘grand desert scenario’).
• Fulfillment of consistency conditions:
– absence of Landau poles up to MP l
– absence of instabilities of effective potential up to MP l
and solution to remaining problems of particle physics,viz. matter/antimatter asymmetry, Dark Matter.
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Conformal Invariance and Quantum Theory
Important Fact: classical conformal invariance is gener-ically broken by quantum effects (unlike SUSY!) ⇒• Impose anomalous Ward identity
Θµµ =
∑
n
β(n)(g)O(n)(x)
[W. Bardeen, FERMILAB-CONF-95-391-T, FERMILAB-CONF-95-377-T]
and try radiative symmetry breaking a la Coleman-Weinberg ⇒mass generation via conformal anomaly?
• Or: admit soft breaking (=explicit mass terms) asis commonly done for MSSM like models, but insiston cancellation of quadratic divergenes
It is well known that implementation of radiative sym-metry breaking in SM encounters difficulties (pertur-bative stability, spurious vacua, quadratic divergences).
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Coleman-Weinberg Mechanism (1973)
• Idea: spontaneous symmetry breaking by radiativecorrections =⇒ can small mass scales be explainedvia conformal anomaly and effective potential ?
V (ϕ) =λ
4ϕ4 → Veff(ϕ) =
λ
4ϕ4 +
9λ2ϕ4
64π2
[
ln
(
ϕ2
µ2
)
+ C0
]
• But: radiative breaking spurious for pure ϕ4 theory
as is easily seen in terms of RG improved potential
V RGeff =
1
4λ(L)ϕ4 =
λ
4· ϕ4
1− (9λ/16π2)LL ≡ ln
(
ϕ2
µ2
)
• With more scalar fields finding minima and ascer-taining their stability is much more difficult, as thereis no similarly explicit formula for V RG
eff (ϕ1, ϕ2, ...).
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Softly broken conformal symmetry (SBCS)
Assume existence of a UV complete and finite funda-mental theory, such that Λ is a physical cutoff to bekept finite, and impose vanishing of quadratic diver-gences at particular distinguished scale Λ (= MPL?),together with further consistency requirements:
• Bare mass parameters should obey mB(Λ) ≪ MPL .
• There should be neither Landau poles nor instabili-ties for MEW < µ < Λ (manifesting themselves as theunboundedness from below of the effective potentialdepending on the running scalar self-couplings);
• All couplings λR(µ) should remain small (for the per-turbative approach to be applicable and stability ofthe effective potential electroweak minimum).
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Bare vs. renormalized couplings
With cutoff Λ and normalization scale µ we have
λB(µ, λR,Λ) = λR +∞∑
L=1
L∑
ℓ=1
aLℓ λL+1R
(
lnΛ2
µ2
)ℓ
,
so that λB = λR for µ = Λ. Thus, keeping the physicalcutoff Λ finite and fixed we can set
fquad(λB) = 0
NB: this condition would not make sense if Λ → ∞where bare couplings are expected to become singular!
In terms of running couplings this condition becomesscale dependent: [cf. Veltman (1982)]
fquad(λR, µ; Λ) ≡ fquad(λB(µ, λR,Λ)) = 0
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Quadratic divergences in Standard Model[M. Veltman(1982);Y.Hamada,H.Kawai,K.Oda, PRD87(2013)5; D.R.T.Jones,PRD88(2013)098301]
Only one scalar: fquad(λR(µ)) = 0 for µ ≈ 1024GeV ≫ MPL !
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Metastability of electroweak vacuum?[Hamada,Kawai,Oda, PRD87(2013)5; Buttazzo,Degrassi,Giardino,Giudice,Sala,Salvio,Strumia(2013)]
λR(µ) becomes negative for µ > 1010GeV ⇒ instability?
→ might also be relevant to cosmology!
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Particle Content: SM vs. CSM
CSM = minimal extension of SM with two new d.o.f.:
• Right-chiral neutrinos νiR → 48 = 3 × 16 fermions
• One extra ‘sterile’ complex scalar φ(x)
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CSM = minimal extension of SM
[K. Meissner,HN, PLB648(2007)312; Eur.Phys.J. C57(2008)493]
• Start from conformally invariant (and therefore renor-malizable) fermionic Lagrangian L = Lkin + L′
L′ :=(
LiΦY Eij E
j + QiǫΦ∗Y Dij D
j + QiǫΦ∗Y Uij U
j +
+LiǫΦ∗Y νijν
jR + φνiTR CY M
ij νjR + h.c.)
− V (Φ, φ)
• Besides usual SU (2) doublet Φ: complex scalar φ(x)
φ(x) = ϕ(x) exp
(
ia(x)√2µ
)
• No fermion mass terms, all couplings dimensionless
• Y Uij , Y
Eij , Y
Mij real and diagonal: Y M
ij = yNiδij
Y Dij , Y
νij complex→ parametrize family mixing (CKM)
• Neutrino masses from usual seesaw mechanism:Y ν ∼ 10−6 and yN ∼ O(1) ⇒ (Y ν)2/yN ∼ 10−12
⇒ no new large scales needed for light neutrinos.
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Scalar Sector of CSM
Right-chiral neutrinos and one complex scalar ⇒V (Φ, φ) = m2
1Φ†Φ +m2
2|φ|2 + λ1(Φ†Φ)2 + 2λ3(Φ
†Φ)|φ|2 + λ2|φ|4
Thus, two real scalars H ≡ (Φ†Φ)1/2 and ϕ ≡ |φ| and(3 + 1) Goldstone fields: three phases for electroweaksymmetry breaking (BEH effect) and one extra phase(a(x) ≡ Majoron). Two mass eigenstates at minimum
(
hS
)
=
(
cosβ sin β− sin β cos β
)(
H − 〈H〉ϕ− 〈ϕ〉
)
Scalar masses Mh < MS and | tan β| < 0.3 (from existingexperimental bounds if h = SM Higgs-Boson).
NB: Explicit mass terms remain compatible with con-formal symmetry because of SBCS requirement (cf.soft breaking of supersymmetry).
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Quadratic divergences in CSM
Two physical scalars ⇒ two conditions (at one loop)
16π2fquad1 (λ, g, y) = 6λ1 + 2λ3 +
9
4g2w +
3
4g2y − 6y2t
!= 0
16π2fquad2 (λ, g, y) = 4λ2 + 4λ3 −
3∑
i=1
y2Ni
!= 0
• Start from known values of electroweak couplings gy, gw, yt at
µ = MEW and evolve them to µ = MPL.
• Choose λ1, yN and determine λ2 and λ3 from fquadk = 0
• Evolve all couplings back to µ = MEW and check whether all
consistency requirements are satisfied.
• In particular: ensure stability of vacuum by following evolu-
tions λ = λ(µ) for MEW < µ < MPL.
This SBCS condition is what mainly distinguishes CSM
from other extensions of SM with extra scalars →above relations can in principle be tested!
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β-functions at one loopExploit partial cancellations to keep all couplings un-der control up to MPL (but not beyond!):
βgs = − 7g3s (asymptotic freedom)
βyt = yt
{
9
2y2t − 8g2s − 9
4g2w − 17
12g2y
}
(1)
βλ1 = 24λ21 + 4λ2
3 − 3λ1
(
3g2w + g2y − 4y2t)
+
+9
8g4w +
3
4g2wg
2y +
3
8g4y − 6y4t (2)
where β ≡ 16π2β. Thus
• From (1) ⇒ gs must be large for µ ∼ MEW !
• From (2) ⇒ yt must be large for µ ∼ MEW !
• Positive contribution +4λ23 from extra scalar keeps
λ1(µ) > 0 for MEW < µ < MPL ⇒ stability of vacuum!
Remarkably, small correction suffices for this purpose!
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What might be observed
• h decay width is decreased: Γh = cos2 β ΓSM < ΓSM
• S decay width: ΓS = sin2 β ΓSM + · · ·– Main term: narrow resonance (‘shadow Higgs boson’)
– Other terms: decay of heavy scalar into two or three h’s,
and two heavy neutrinos (if kinematically allowed).
• → new scalar boson is main prediction! But maynot be easy to find... e.g. if β is small.
• Decay via two h bosons might produce spectacularsignatures with 5,...,8 leptons coming out of a singlevertex. But: rates vs. background?
• Proposal can be discriminated against other exten-sions of SM that produce similar signatures, butwould come with extra baggage (e.g. charged scalarsas in SUSY or two doublet models).
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CP Violation and Leptogenesis
In usual SM, CKM matrix only source of CP violationvia electroweak interactions → known to be too smallto explain observed matter/antimatter asymmetry:
YB ≡(
nB
nγ
)
obs
= 10−10 vs.
(
nB
nγ
)
SM
= 10−14
In CSM the Yukawa coupling matrix Y νij provides extra
source of CP breaking. Assume Y Mij = yMδij with M ≡
yM〈ϕ〉 and parametrize [Casas-Ibarra(2001)]
m ≡ 〈H〉Y ν = M 1/2U [diag(m1/21 , m
1/22 , m
1/23 )] RT
with R ∈ SO(3,C) and U ∈ U (3) (9 + 3 + 6 = 18 param-eters) ⇒ seesaw-type mass matrix for light neutrinos
mM−1mT = U∗ [diag(m1, m2,m3)] U
Thus U = neutrino analog of CKM matrix (“PMNSmatrix”), but with three phases. [Pontecorvo;Maki,Nakagawa,Sakata]
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Thus R plays no role for measurable low energy pa-rameters (neutrino oscillations,...) but does enter Y ν
ij
which is also responsible for CP violating decays ofheavy neutrinos, and thus for leptogenesis.
Heavy neutrino masses are O(1 TeV) ⇒ must invokeresonant leptogenesis (which requires nearly degener-ate heavy neutrino masses) [Pilaftsis(1997);Pilaftsis,Underwood(2004)]
For heavy neutrino, small mass splittings are inducedby Y ν
ij (breaks SO(3) symmetry of Majorana mass term)
(MN)ij = Mδij +1
2M−1[(m∗mT )ij + c.c.] + · · ·
Thus can play with R ∈ SO(3,C) in order to get YB
of desired order of magnitude without affecting low
energy neutrino physics. [→ A.Lewandowski]
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Table 1: Some exemplary values [cf. PRD97(2018)035024]; always Mh = 125.09 GeV
MS[GeV] sin β MN [GeV] 〈ϕ〉[GeV] YB Γh[MeV] ΓS[GeV] Br(S → [SM]) Br(S → hh)
1030 -0.067 1604 17090 7.9 ×10−11 4.19 4.02 0.78 0.2893 -0.076 1238 11331 1.2×10−10 4.186 3.3 0.76 0.2839 -0.082 1181 11056 1.2×10−10 4.182 3.08 0.76 0.2738 -0.093 1052 10082 1.1×10−10 4.174 2.66 0.76 0.22642 -0.11 1303 19467 1.7×10−10 4.16 2.34 0.76 0.22531 -0.13 949 12358 1.5×10−10 4.138 1.92 0.74 0.22393 -0.18 801 12591 1.0×10−10 4.07 1.28 0.72 0.26362 -0.20 815 14534 1.6×10−10 4.04 1.06 0.68 0.3350 -0.21 738 12302 7.4×10−11 4.028 0.96 0.66 0.32320 -0.23 751 14437 1.3×10−10 3.984 0.86 0.66 0.32279 -0.28 683 14334 9.6×10−11 3.896 0.68 0.7 0.28258 -0.31 675 15752 1.3×10−10 3.824 0.54 0.78 0.20
• Measured value Γh = (4.07 ± 0.2)MeV
• MS < MN ⇒ no decay S → Nν possible
• Without suppression factor sin2 β would get ΓS ∼ 1 TeV
• Can arrange for YB ∼ 10−10 → leptogenesis.
→ CSM can be made compatible with current data!
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Majoron as DM candidate?
Because φ → e−2iωφ under lepton number symmetry,non-vanishing vev 〈φ〉 6= 0 breaks lepton number sym-metry spontaneously ⇒ Goldstone boson ≡ Majoron.
Because of cancellation of (B−L) anomalies this Gold-stone boson remains massless to all orders in pertur-
bation theory in flat space QFT → two options:
• Gauge U (1)B−L ⇒ new Z ′ vector boson? [cf. Iso,Okada,Orikasa(2009)]
• Don’t gauge → exactly massless scalar particle?
‘Folklore Theorem’: in quantum gravity there cannotexist unbroken continuous global symmetries → Con-jecture: tiny mass generated by quantum gravity
L ∼v4φM 2
PL
φ2 + c.c. ⇒ mGoldstone ∼ 10−3eV
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Outlook
• Usual SM probably cannot survive to Planck scale⇒ requires some extension, if only to accommodateright-chiral neutrinos and to recover stability.
• A conformally motivated minimal extension of SMcan in principle satisfy all consistency requirements.
• Proper embedding into a UV complete theory?
• Low energy supersymmetry may after all not berequired for stability of electroweak scale...
• ... but is probably needed for a UV complete theoryof quantum gravity and quantum space-time.
• Nevertheless: because of its minimality CSM maybe difficult to generate from string theory.
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Outlook
Conclusion: Nature is probably still
a bit smarter than we think, and may
have a few more tricks up her sleeve!
THANK YOU
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Conformal invariance from gravity?
Here not from scale (Weyl) invariant gravity, but:
N = 4 supergravity 1[2]⊕ 4[32]⊕ 6[1]⊕ 4[1
2]⊕ 2[0]
coupled to n vector multiplets n× {1[1]⊕ 4[12]⊕ 6[0]}Gauged N = 4 SUGRA: [Bergshoeff,Koh,Sezgin; de Roo,Wagemans (1985)]
• Scalars φ(x) = exp(LIAT I
A) ∈ SO(6, n)/SO(6)× SO(n)
• YM gauge group GYM ⊂ SO(6, n) with dim GYM = n + 6
[Example inspired by ‘Groningen derivation’ of conformal M2
brane (‘BGL’, ‘ABJM’) theories from gauged D = 3 SUGRAs]
Although this theory is not conformally invariant, theconformally invariant N = 4 SUSY YM theory nev-ertheless emerges as a κ → 0 limit, which ‘flattens’spacetime (with gµν = ηµν + κhµν) and coset space
SO(6, n)/((SO(6)× SO(n)) −→ R6n ∋ φ[ij] a(x)
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However: conformality of limit requires extra restric-tions, in particular compact gauge group:
GYM ⊂ SO(n) ⊂ SO(6, n)
Exemplify this claim for scalar potential: with
Caij = κ2fabcφ[ik]
bφ[jk] c +O(κ3) , Cij = κ3fabcφ[ik]aφb [kl]φ[lj]
c +O(κ4)
potential of gauged theory is (m,n = 1, . . . , 6; κ|z| < 1)
V (φ) =1
κ4
(1− κz)(1− κz∗)
1− κ2zz∗
(
CaijCai
j −4
9C ijCij
)
= Tr [Xm, Xn]2 +O(κ)
Idem for all other terms in Lagrangian! Unfortunately
• N = 4 SYM is quantum mechanically conformal theory
→ no conformal anomaly → no symmetry breaking!
• Thus need non-supersymmetric vacuum with Λ = 0
⇒ must look for a better theory with above features!
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Metamorphosis of CW mechanism?
But: if we embed SM in a UV finite theory of quantum
gravity what is the origin of (conformal) anomalies?
Conjecture: If this (unknown) theory admits a clas-sically conformal flat space limit, CW-like contribu-tions could arise from finite logarithmic (in κ) quan-
tum gravitational corrections ⇒ identify v ∼ MPl!
• CW-like corrections would not be due to UV diver-gences, but rather to the fact that gravity is not
conformal – this would be the only ‘footprint’ thatquantum gravity leaves in low energy physics.
• Observed mass spectrum and couplings in the SMcould have their origin in quantum gravity.
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CSM = minimal extension of SM
[K. Meissner,HN, PLB648(2007)312; Eur.Phys.J. C57(2008)493]
• Start from conformally invariant (and therefore renor-malizable) fermionic Lagrangian L = Lkin + L′
L′ :=(
LiΦY Eij E
j + QiǫΦ∗Y Dij D
j + QiǫΦ∗Y Uij U
j +
+LiǫΦ∗Y νijν
jR + φνiTR CY M
ij νjR + h.c.)
− V (Φ, φ)
Recall that SM contains three electroweak SU(2)w doublet fermions:
Qi ≡(
uiαdiα
)
, Li ≡(
νiαeiα
)
and electroweak singlets U iα , D
iα , E
iα (where i, j, ... = 1, 2, 3 label the
three families). In addition, CSM contains three spinors N iα (→
right chiral neutrinos). Use Weyl spinors such that, for instance,
ΨiE =
(
eiαEiα
α
)
, ΨiU =
(
uiαU iα
)
, Ψiν =
(
νiαN iα
)
≡ νiL + νiR , etc.