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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

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?

A basic question (Ockham’s razor)

What is the minimal extension required tokeep the SM viable up to MPL?

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);....]

• ......

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.

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!

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.

Low energy supersymmetry?

Low energy exotics?

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).

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.

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.

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).

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, ...).

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).

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

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 !

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!

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)

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.

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).

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!

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!

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).

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]

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]

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!

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

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.

Outlook

Conclusion: Nature is probably still

a bit smarter than we think, and may

have a few more tricks up her sleeve!

THANK YOU

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)

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!

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.

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.