Schmöckwitz, 28 August Hermann Nicolai MPI für Gravitationsphysik, Potsdam (Albert Einstein Institut)

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1 bla Quadratic Divergences and the Standard Model Schmöckwitz, 28 August 2014 Hermann Nicolai MPI für Gravitationsphysik, Potsdam (Albert Einstein Institut) Based on joint work with Piotr Chankowski, Adrian Lewandowski and Krzysztof Meissner, arxiv: For related work, see Antipin, Mojaza and Sannino, arxiv:

2 The electroweak hierarchy problem A main focus of BSM model building over many years: m 2 R = m 2 B +δm 2 B, δm 2 B Λ 2 and m 2 R Λ2 But is this really a problem? Not in renormalized perturbation theory because Λ and because renormalisation does not care whether an infinity is quadratic or logarithmic! (as exemplified by dimensional regularisation which does not even see quadratic divergences for d = 4+ε). Yes, if SM is embedded into Planck scale theory and Λ is a physical scale (cutoff) quadratic dependencies on cutoff imply extreme sensitivity of low energy physics to Planck scale physics.

3 Two popular proposed solutions Low energy supersymmetry: exact cancellation of quadratic divergences by (softly broken) supersymmetry choice of cutoff Λ does not matter, can formally send Λ and adopt any convenient renormalisation scheme. Technicolor (motivated by QCD): no fundamental scalars no quadratic divergences H boson would have to be composite (could still be true...)... as well as a number of other ideas NB: these proposals would only solve the technical part of the hierarchy problem (= stabilising small numbers against large perturbative corrections), but would not explain the observed hierarchy of scales!

4 Low energy supersymmetry?

5 Low energy exotics?

6 Absence of any evidence (so far) from LHC for either of these options explore alternative options Can the SM survive all the way to Planck scale M PL? In this talk: explore softly broken conformal symmetry (SBCS) for a minimal extension of usual SM as an alternative option. NB: this proposal does without low energy supersymmetry, but supersymmetry is probably still essential for a finite and consistent theory of quantum gravity. Realisation of such a scenario would move the SUSY breaking scale back up to the Planck scale, but make no explicit assumptions about Planck scale theory other than its UV finiteness (= UV completeness).

7 Reminder: the conformal group SO(2, 4) This is an old subject! [see e.g. H.Kastrup, arxiv: ] Conformal group = extension of Poincaré group (with generators M µν,p µ ) by five more generators D and K µ : Dilatations (D) : x µ e α x µ Special conformal transformations (K µ ): x µ = xµ x 2 c µ 1 2c x+c 2 x 2 e iαd P µ P µ e iαd = e 2α P µ P µ 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 as an exact symmetry in nature.

8 Conformal Invariance and the Standard Model Fact: Standard Model of elementary particle physics is conformally invariant at tree level except for explicit mass term m 2 Φ 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 consistency conditions(absence of Landau poles and of instabilities of effective potential), and absence of any intermediate mass scales between M EW and M PL ( grand desert scenario ).

9 Evidence for large scales other than M Pl? (SUSY?) Grand Unification: m X O(10 15 GeV)? But: proton refuses to decay (so far, at least!) SUSY GUTs: unification of gauge couplings at O(10 16 GeV) Light neutrinos (m ν O(1eV)) and heavy neutrinos most popular (and most plausible) explanation of observed mass patterns via seesaw mechanism: [Gell-Mann,Ramond,Slansky; Minkowski; Yanagida] m (1) ν m2 D M, m D = O(m W ) m (2) ν M O(10 12 GeV)? Resolution of strong CP problem need axion a(x). Limits e.g. from axion cooling in stars L = 1 af µν F µν with f a O(10 10 GeV) 4f a NB: axion is (still) an attractive CDM candidate.

10 Conformal Invariance and Quantum Theory Important Fact: classical conformal invariance is generically broken by quantum effects (unlike SUSY!) Impose anomalous Ward identity Θ µ µ = n β (n) (g)o (n) (x) [W. Bardeen, FERMILAB-CONF T, FERMILAB-CONF T] and try radiative symmetry breaking à la Coleman- Weinberg. But: quadratic divergences? Admit soft breaking (=explicit mass terms) as is commonly done for MSSM like models, but insist on cancellation of quadratic divergenes NB: it is known that option (1) does not work for usual SM with one physical Higgs, but with one extra complex scalar (as in our model) there is more freedom.

11 SBCS Consistency Requirements Assume existence of a UV complete and finite fundamental theory, such that Λ is a physical cutoff to be kept finite, and impose vanishing of quadratic divergences at particular distinguished scale Λ (= M PL?) : Bare mass parameters should obey m B (Λ) M PL ; there should be neither Landau poles nor instabilities for M EW < µ < Λ (manifesting themselves as the unboundedness from below of the effective potential depending on the running scalar self-couplings); all couplings λ R (µ) should remain small (for the perturbative approach to be applicable and stability of the effective potential electroweak minimum). Furthermore use known SM values of couplings and masses as input parameters at µ = M EW.

12 Bare vs. renormalized couplings With cutoff Λ and normalization scale µ we have λ B (µ,λ R,Λ) = λ R + so that λ B = λ R for µ = Λ, and L=1 L l=1 a Ll λ L+1 R ) l (ln Λ2, µ 2 L (ln Λ2 ) l m 2 B (µ,λ R,m R,Λ) = m 2 R ˆf quad (µ,λ R,Λ)Λ 2 + m 2 R L=1 l=1 c Ll λ L R µ 2 Crucial fact: coefficient of Λ 2 can be written as a function of the bare coupling(s) only, i.e. ˆfquad (µ,λ R,Λ) f quad (λ B (µ,λ R,Λ)). Thus, keeping the physical cutoff Λ finite we can set f quad (λ B ) = 0 NB: this condition would not make sense if Λ where bare couplings are expected to become singular!

13 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: f quad (λ R (µ)) = 0 for µ GeV M PL!

14 Is the Standard Model doomed? [Y.Hamada,H.Kawai,K.Oda, PRD87(2013)5] λ R (µ) becomes negative for µ > GeV instability? might also be relevant to cosmology!

15 Minimal extension of SM = CSM [K. Meissner,HN, PLB648(2007)312; Eur.Phys.J. C57(2008)493] Start from conformally invariant(and therefore renormalizable) fermionic Lagrangian L = L kin +L L := ( L i ΦYij E E j + Q i ǫφ Yij D D j + Q i ǫφ Yij U U j + + L i ǫφ Yijν ν j R +φνit R CYij M ν j R +h.c.) V(Φ,φ) Besides usual SU(2) doublet Φ: new scalar field φ(x) φ(x) = ϕ(x)exp ( ) ia(x) 2µ No fermion mass terms, all couplings dimensionless Yij U, Y ij E, Y ij M real and diagonal: Yij M = y Ni δ ij Yij D, Y ij ν complex parametrize family mixing (CKM) Neutrino masses from usual seesaw mechanism (but with φ < O(1TeV) no new large scale)

16 Scalar Sector of CSM Right-chiral neutrinos and one complex scalar V(Φ,φ) = m H Φ Φ+m 2 φ φ 2 +λ 1 (Φ Φ) 2 +2λ 3 (Φ Φ) φ 2 +λ 2 φ 4 where Φ = (Φ 1,Φ 2 ) is the SU(2) EW doublet and φ is the extra gauge singlet. At the minimum 2 Φi = v H δ i2, 2 φ = vφ with mass eigenstates h 0 and ϕ 0 ( h 0 ϕ 0 ) = ( cosβ sinβ sinβ cosβ )( ) 2Re(Φ2 Φ 2 ), (1) 2Re(φ φ ) with masses M h < M ϕ and tanβ < 0.3 (as dictated by existing experimental bounds if h 0 = SM Higgs- Boson).

17 Quadratic divergences in CSM Two physical scalars two conditions (at one loop) 16π 2 f quad 1 (λ,g,y) = 6λ 1 +2λ g2 w g2 y 6yt π 2 f quad 2 (λ,g,y) = 4λ 2 +4λ 3 Start from known values of electroweak couplings g y,g w,y t at µ = M EW and evolve them to µ = M PL. Choose λ 1,y N and determine λ 2 and λ 3 from f quad k = 0 Evolve all couplings back to µ = M EW and check whether all consistency requirements are satisfied. leads to a range of possible values for new heavy scalar ϕ 0 and heavy neutrinos (with m N < 1TeV). i=1 y 2 N i

18 β-functions at one loop β (1) λ 1 = 24λ λ2 3 3λ 1( 3g 2 w +g 2 y 4y2 t) g4 w g2 wgy g4 y 6yt β (1) λ 2 = 20λ λ2 3 +2λ 2 y 2 y 4 N i N i i=1 i=1 { β (1) λ 3 = 1 2 λ 3 24λ 1 +16λ 2 +16λ 3 ( } ) 3 9gw 2 +3gy 2 +2 y 2 +12y 2 N i t β g (1) w = 19 6 g3 w, β(1) g y = 41 6 g3 y, β(1) g s = 7gs 3 {, 9 β y (1) t = y t 2 y2 t 8gs g2 w 17 }, β (1) y Nj = 1 2 y N j {2y 2 N j + where β 16π 2 β. } 3 y 2 N i i=1 12 g2 y i=1

19 Admissible parameter ranges Coupling Λ 1, Λ 2, Λ 3, y N 2 Mφ TeV t Β t Β t Β log 10 Μ M EW m N TeV Couplings remain small (also at two loops) M ϕ grows with decreasing mixing angle β Usual seesaw mechanism applies, such that Small light neutrino masses with Y ν Heavy neutrinos not extremely heavy: m N < 1TeV? Caveats: scheme dependencies, threshold effects?

20 What might be observed h 0 decay width is decreased: Γ h 0 = cos 2 βγ SM < Γ SM Crucial relation for ϕ 0 decay width: Γ ϕ 0 = sin 2 βγ SM + Γ ϕ 0 h 0 h 0 + Γ ϕ 0 ν R ν R First term: narrow resonance ( shadow Higgs boson ) Second and third terms: decay of heavy scalar into two or three h 0 s, and two heavy neutrinos (if kinematically allowed, i.e. M ϕ > 2m N for at least one species). Decay via two h 0 bosons might produce spectacular signatures with 5,...,8 leptons coming out of a single vertex. But: rates vs. background? Proposal can be easily discriminated against other extensions of SM that might produce similar signatures, but that would come with a lot of extra baggage (accompanying signatures).

21 Outlook Usual SM probably cannot survive to Planck scale requires some extension, if only to accommodate right-chiral neutrinos. A conformally motivated extension of the SM can in principle satisfy all consistency requirements, if properly embedded into a UV complete theory. Model accommodates axionsnaturally: f a m 2 W /m ν. Low energy supersymmetry may after all not be required for stability of electroweak scale but is probably needed for a UV complete theory of quantum gravity and quantum space-time. Conclusion: Nature is probably still a bit smarter than us, and may have a few more tricks up her sleeve!