Implications of LHC unobservations

Theory Experiment Naturalness? Agravity Inflation E → ∞

Alessandro Strumia Talk at Galileo Galilei Institute, October 16, 2015 Inspired by the ghost of Galileo Theory The hierarchy problem

−40 2 −32 2 Why Fgravity ∼ 10 Felectric? In the SM because MHiggs ∼ 10 MPl.

But no symmetry forces this smalness, and quantum corrections seem large

12λ2 ( 1 2 2(top) = topΛ2 δMh ≈ δMh ≈ 2 × 2 2 (4π) ln MPl/Λ

√ ( 400 GeV δM2 <∆ × M2 needs new physics below Λ < ∆ × h ∼ h ∼ 50 GeV “light fundamental scalars do not exist, unless they are accompanied by new physics that protects their lightness from quadratically divergent corrections”

. Past performance

√ Those Higgs-like scalars present in field theories of condensed matter are not unnaturally lighter than their ultimate cut-off: the atomic lattice. √ The electron mass receives divergent electromagnetic corrections Naturalness holds thanks to new physics: chiral symmetry of e± fermions. √ m2 − m2 receives power divergent electromagnetic corrections. π± π0 Naturalness holds thanks to new physics: π are QCD composite of fermions. √ K mixing receives power divergent corrections. Naturalness holds thanks to new physics: the charm. The solution to the hierarchy problem SUSY is an entirely theoretical idea – there is no experimental evidence for it … yet → talk by Mihoko Nojiri

The ? SUSY stabilizes Higgs: the weak scale Mrs. SUSY If weak-scale SUSY existed, it could … scalar is the scale of SUSY breaking. precipice ? SUSY• Moderate extends the hierarchy Lorentz, problem allows spin 3/2. by cancelling quadratic SUSY unifies fermions with bosons. ? divergence of SM scalar ? SUSY unifies gauge couplings. EW scale—1 • Equalise the number of fermionic and bosonic degrees of freedom, render existence of scalar particles natural

• Realise grand unification of the Mr. Higgs gauge couplings

• Provide a suitable dark matter candidate

GUT scale—1 ? SUSY gives DM aka ‘neutralino’. ? SUSY is predicted by super-strings. Fundamental scalar length scale ? Worry: too many sparticles at LHC? Lepton-Photon, 24–29 June, 2013 Andreas Hoecker — Searches for Supersymmetry at Colliders 2

Experiment LHC found the Higgs

Mh = 125.15 GeV 4 7 8 7 8 7 8 7 8 8 8 7 8 8 7 8 7 8 8 7 7 8 7 8 8 7 7 8 7 8 7 8 7 8 7 8 8 8 8 8 7 8 7 8 7 8 7 8 D0 D0 D0 D0 - - - - Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas CMS CMS CMS CMS CMS CMS CMS CMS Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas Atlas CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS CMS 3 CDF CDF CDF CDF

2 rate SM 

Rate 1 l l l 0 0 1 1 2 - - - T - - - - - l jj L L E jj jj jj jj jj mj - j - { T T ------0 0 1 1 2 2 3 3 4 tight ct ct bb 2 ΓΓ 1 loose loose crl crl url url E E ccl ccl jj ucl ucl crh crh ttH ΤΤ ΤΤ urh urh cch cch uch uch V inv inv Γ Γ ------lm2j - - - - - ttH - - lhm2j thm2j ------® - - ® - - - - - ΤΤ

0 - - ΜΜ ΜΜ VH - - ΓΓ Z Z ZZ ZZ ® ® ttH - VH ΤΤ ΤΤ bbV bbV bbV WW WW WW h - h - - Zh Zh ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ h h - H ΓΓ ΓΓ H ΓΓ - ΓΓ VBF VBF VBF VBF VBF ΓΓ ΓΓ - ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ VH VH VH tt ΓΓ tt ΓΓ - - - - - ΓΓ ΓΓ ΓΓ - ΓΓ - - ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ ΓΓ

-1 The SM Higgs

Fit to Higgs couplings

1

0.3

prediction coupling 0.1 SM

Higgs Τ b W Z t 0.03

0.01

1 3 10 30 100 300 Mass of SM particles in GeV

[Giardino, Kannike, Masina, Raidal Strumia, 1303.3570] ' official And nothing else Maybe up to the Planck scale

For the measured Mh, Mt the SM can be extrapolated up to MPl.

1.0 0.10

0.08 3Σ bands in yt g3 0.8 Mt = 173.3 ± 0.8 GeV HgrayL Α3HMZL = 0.1184 ± 0.0007HredL 0.06 Λ Mh = 125.1 ± 0.2 GeV HblueL 0.6 g2 0.04 coupling g1 couplings 0.4 0.02 quartic

SM Mt = 171.1 GeV

0.2 Higgs 0.00 m in TeV ΑsHMZL = 0.1205 Λ -0.02 ΑsHMZL = 0.1163 0.0 yb Mt = 175.6 GeV -0.04 102 104 106 108 1010 1012 1014 1016 1018 1020 102 104 106 108 1010 1012 1014 1016 1018 1020 RGE scale Μ in GeV RGE scale Μ in GeV

λ and its β-function nearly vanish around MPl (!!?) Data favour λ < 0: unstable SM potential

180 107 108 Instability 109 178 1010 1011 176 12 GeV 10 13

in 10 t

M 1016 174

mass 1,2,3 Σ Meta-stability pole 172 1019 Top

170 1018 1014 Stability 168 120 122 124 126 128 130 132 1017 Higgs pole mass Mh in GeV Not a problem: vacuum decay is slow enough; cosmological Higgstory can naturally end up in the small SM minimum. Extrapolating the SM — Technical issues — Computing the SM parameters

SM parameters extracted from data at 2 loop accuracy: atµ ¯ = Mt   Mt MW − 80.384 GeV g2 = 0.64822 + 0.00004 − 173.10 + 0.00011 GeV 0.014 GeV  M  M − 80.384 GeV g = 0.35761 + 0.00011 t − 173.10 − 0.00021 W Y GeV 0.014 GeV   Mt α3(MZ) − 0.1184 yt = 0.9356 + 0.0055 − 173.10 − 0.0004 ± 0.0005th GeV 0.0007  M   M  λ = 0.1271 + 0.0021 h − 125.66 − 0.00004 t − 173.10 ± 0.0003 . GeV GeV th m  M   M  = 132.03 + 0.94 h − 125.66 + 0.17 t − 173.10 ± 0.15 . GeV GeV GeV th Renormalization to large energies done with 3 loop RGE.

[Buttazzo, Degrassi, Giardino, Giudice, Sala, Salvio, Strumia, 1307.3536] The top mass is the main uncertainty

• Mt = 173.34 ± 0.76 GeV from LHC, TeVatron. However: i) the pole mass is defined up to ±O(ΛQCD) ∼ ±0.3 GeV; ii) it is a MonteCarlo mass: like reconstructing the pig mass from sausages. • Mt = 174.6 ± 1.9 GeV from top production cross sections. • Mt = 177.0 ± 2.6 GeV from EW data, dominated by MW (LHC will measure it 2-3 times better). Observables with a quadratic dependence on Mt:

3 4 y yt 2 + - t DΡ µ y AHZ®bb, Bs®Μ Μ , K®ΠΝΝL µ DmB, ΕK µ t 2 Mt Mt

d d t d

t

¶Μ Χ ¶Μ Χ ¶Μ Χ Χ Χ Χ

d d t d

• Mt = 173.9 ± 7.8 GeV from flavor (∆mBs). Uncertainty will decrease down + − to 1.7 GeV (2025) thanks to Bs → µ µ . • A linear collider at the top threshold would measure Mt up to ±0.05 GeV. [Giudice, Paradisi, Strumia, 1508.05332] Computing vacuum decay in the SM

The vacuum decay rate is suppressed by the action of the tunnelling bounce s 2 2 2R 2 2 2 Z 4 8π 1 h(r) = r = ~x − t ,S = d x LSM(h(r)) = . |λ|r2 + R2 3 |λ|

Our vacuum is unstable (decay rate faster than the age of the universe TU ) if 1 S <500 ∼ ln M T ⇒ λ < − 0.05 ∼ 4 Pl U ∼ At tree level R is arbitrary and λ is constant. At loop level λ runs, S → Seff which roughly means: use λ renormalized at ∼ 1/R. For Mh ≈ 115 GeV the vacuum decay bound is saturated and 1/R ≈ 1017 GeV.

0.15 [Isidori, Ridolfi, Strumia, hep-ph/0104016] ? Computable gravitational corrections are small. 0.10 potential 0.05 SM potential SM .

[Isidori, Rychkov, Strumia, Tetradis, 0712.0242] 4 B h  V 4 0.00 For 125 GeV, 1 gets close to , but = M ≈ /R MPl eff h Λ Slow enough vacuum decay S  500. Of course, new physics can add new -0.05 decay channels (extra scalars; string landscape ... -0.10 Too fast vacuum decay

Branchina emphasizes a out-of-control example) -0.15 102 104 106 108 1010 1012 1014 1016 1018 1020 h vev in GeV The SM effective potential

Recent claims in the literature: ‘Be consistent’ [Schwartz et al.], ‘resum IR-enhanced effects’ [Espinosa et al.] Right, but in Landau gauge ξ = 0 these subtelties start only at ≥ 3 loops. ‘The instability scale is badly gauge dependent’ [diLuzio, Mihaila]

The effective potential depends on gauge parameters ξ. Nielsen proofed that ∂S some ! δS d + × = 0 i.e. S[h (h), ξ] = 0 ∂ξ kernel δh dξ ξ where S[h, ξ] is the quantum effective action, with the local expansion:

Z 4 2 S = d x[V (h) + K(h)(∂µh) + ··· ] Implications of Nielsen theorem

• Vmax and Vmin and the stability condition ∆V = 0 are gauge-independent. • The bounce action is gauge-independent, because it is a classical solution. • The classical e.o.m. is gauge-independent, at least at leading order in ∂. • The same for Langevin and Fokker-Planck equations (used later). • Redefine h → h0 making the higgs kinetic term canonical: the whole V (h) becomes gauge-independent if one only considers log-enhanced corrections:

´ 38 0.10 5 10

SM potential in different Ξ gauges 38 0.08 4´10 Dashed: non-canonical Higgs Continous: canonical higgs field -4 0.06 3´1038 -2

4 0 h

 1 L 4 2 h 0.04 ´ 38 H 2 10 4 GeV V 4 in

L 10 = h

0.02 H 1´1038 V eff Λ 0.00 4 10 0 0 1 2 -2 hmax Ξ = -6 -4 -0.02 -1´1038 -0.04 104 106 108 1010 1012 1014 1016 1018 1020 -2´1038 0 2´1010 4´1010 6´1010 Higgs h in GeV Higgs h in GeV

• The gauge dependence of the instability scale is now negligible. My con- jecture: “the gauge dependence of S[h] is just a local reparametrization in field space: all reparametrization-independent content of S is physical”. Cosmological Higgstory Illustrative

→

If your mexican hat turns out to be a dog bowl you have a problem... Higgs mass during inflation

1 2 2 During inflation with Hubble H, the Higgs potential becomes V ≈ VSM+2m h .

Various sources can contribute to extra inflationary Higgs mass m:

1. the inflaton vev φ in presence of a quartic λh2φ2: m2 = 2λφ2; < 2. thermal effects due to inflaton decay, m∼ H; 2 2 infl 3. the gravitational coupling −ξHh R/2: m = ξHR = −12ξHH.

0.10

0.05 The SM RGE running unavoidably in- duces |ξ | >0.01 0.00 H ∼ L Μ H -0.05 H

1 2 ! Ξ dξH ξH + 6 2 9 2 g1 = 6y − (g + ) + 12λ +···-0.10 d lnµ ¯ (4π)2 t 2 2 5 H So later we focus on 3. -0.15 -0.20 104 106 108 1010 1012 1014 1016 1018 1020 Renormalization scale Μ in GeV Higgs inflationary fluctuations

3 If m < 2H, the higgs fluctuates as approximated by the Langevin equation: dh 1 dV (h) H3 + = η(t), hη(t)η(t0)i = δ(t − t0). dt 3H dh 4π2

104 Neglecting V , after N e-folds one gets a Gaussian with variance 102 q 2 H √ hh i = N. -3 N -3 N 2π pHÈhÈ>hmaxL > e pHÈhÈ®¥L > e

max 0 Space-time splits into three regions: h 10  H • h < hmax up to the end of inflation. • h is fluctuating above hmax at the 10-2 end of inflation. No Higgs fluctations above the barrier • h falls to the true deep (Planckian?) minimum. 10-4 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

Higgs coupling to gravity ΞH Are red bubbles bad?

Do regions where the higgs fall to h ∼ MPl expand? Eating all the space? - Of course YES, because things fall to lower energy. 4 - Of course NO, because negative V ∼ −MPl gives AdS that contracts To clarify, we performed a general relativistic computation in thin-wall approx- imation with surface energy density σ.

 2 2  2 dr 2 2 r  −f (r) dη + + r dΩ , r < R f = 1 +  in 2 in 2  fin(r) `in 2  2 2 2 ds = −dτ + R (τ)dΩ2 wall  2 2  dr r 2GM  ( ) 2 + + 2 Ω2 = 1  −fout r dt r d 2, r > R fout − 2 −  fout(r) `out r Result: equation of motion for R(τ) in some ‘effective potential’ 2 1 ± cte R3! cte0 V (R) = − − cte00 R2 ?R2 R Red bubbles are bad

2 The volume energy is controlled by∆ ≡ Vout − Vin − (4πGσ) : its last term is the Newton-like gravitational energy of the surface.

Abnormal case ∆ < 0. Red bubbles, if big enough, expand safely behind a black hole horizon. Can they be DM? No, because of iso-curvatures.

r = 0 r = ¥ r = 0 r ¥ = IIa IIb = II ¥ r 0 ¥ = = III I r r IV I IV r = ¥ ¥ IV = r r = 0 IIIa IIIb r = 0 r = ¥

Normal case: ∆ > 0. Red bubbles expand, if big enough. For an infinite time If seen from inside (irrelevant) they contract and collapse (Cauchy singularity) in a finite time. During inflation they make Sen cheese. After they eat Minkowski.

Conclusion: we must impose no red region in our past light-cone. Are orange bubbles good? More likely yes

Regions with h > hmax at the end of inflation can fall to h = 0 or fall to h ∼ MPl.

Minimal scenario:

• the inflaton releases its energy during a matter-dominated epoch until t ∼ 2 3/2 1/Γφ. In this phase m = −3ξHH/a .

q • later the radiation-dominated epoch starts with T ∼ TRH ∼ MPlΓφ. The Higgs acquires a thermal contribution m2 ∼ +T 2, which is typically large, 3 2 even if the inflaton decays gravitationally, Γφ ∼ Mφ /MPl. > Thermal fluctuations do not induce thermal vacuum decay for Mh∼ 120 GeV.

Rough conclusion: orange bubbles tend to fall towards h = 0. Final result

H can be orders of magnitude above he instability scale hmax

104

102 -3 N -3 N pHÈhÈ>hmaxL > e pHÈhÈ®¥L > e max 0 h 10  H 10-2 No Higgs fluctations above the barrier 10-4 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

Higgs coupling to gravity ΞH 13 2 Even if hmax is small, tensor modes can be observable, r = 0.1(H/8 10 GeV) . A model-dependent Higgs/inflaton quartic can make constraints even weaker. [R.U.M.A. collaboration, 1505.04825] Meta-stable is good?

Today Λ > 0: some theorists argue that de Sitter is inconsistent and must decay. The SM Higgs instability offers this way out, in a restricted range:

180 6 8 10 12 14 16 TRH = 10 GeV 10 10 10 1010 : bad GeV = 0 at T in

t 175 Unstable M bad? > 0: at T mass 170 Unstable

pole dS Stable dS: bad? stable

Top - Meta 165 100 110 120 130 140

Higgs pole mass Mh in GeV

171 GeV < Mt < 175 GeV So... LHC inverse problem solved

SUSY naturalness in trouble Naturalness in trouble

Smart attempts of adding more new physics to explain why we see nothing. [(Baroqueness, Strangeness) plane introduced by A. Falkowski] Ant**pic selection in a multiverse?

−120 4 Vmin ≈ 10 MPl poses another hierarchy problem that no theory can explain, except anthropic selection in a multiverse of > 10120 vacua.

In the SM (yd − yu)v ≈ αemΛQCD: raising v destroys chemistry.

Is Mh  MPl just another anthropic fine tuning? Probably no. Even if we live in a multiverse, there is no need for a fine-tuning 2 2 as unlikely as v /MPl. The most likely outcome should have been: • one of the existing natural theories, like SUSY; • or an anthropically acceptable alternative to the SM that does not involve an unnaturally light Higgs scalar; • or (even within the Standard Model) a smaller y or a smaller MPl. Subtle is the Lord

A century ago the æther was the paradigm, so dragging it become mainstream theory even after the Michelson-Morley experiments. Until relativity emerged. Today data tells: SM, SM, SM. Theory tells: BSM, BSM, BSM. The contradiction is confusing us, but nature is surely following some logic

Natural solution: Napoli = Salerno. But not supported by geo data Anthropic solution: mafia sells signposts. Plausible but untestable Or think different Redefining naturalness

[Caution: this is when rotten tomatoes start to fly] Is nature natural?

The goal of this talk is presenting an alternative: a renormalizable theory valid above MPl such that Mh is naturally smaller than MPl without new physics at the weak scale. It naturally gives inflation and an anti-graviton ghost-like.

Quadratic divergences vanish and a modified Finite Naturalness applies.

Power divergences and regulators are suggested by QFT equations. The æther was suggested by Maxwell equations. But they are unphysical. Maybe we are again over-interpreting. Maybe power divergences vanish because the ultimate unknown physical cut-off behaves like dimensional regularization. Or maybe there are no regulators: a SM-like theory holds up to infinite energy.

[Pappadopulo Farina, Strumia, 1303.7244] Ipse undixt

Wilson proposed the usual naturalness attributing a physical meaning to mo- mentum shells of power-divergent loop integrals, used in the ‘averaged action’.

“The final blunder was a claim that scalar elementary particles were unlikely to occur in elementary particle physics at currently measurable energies unless they were associated with some kind of broken symmetry. The claim was that, otherwise, their masses were likely to be far higher than could be detected. The claim was that it would be unnatural for such parti- cles to have masses small enough to be detectable soon.

But this claim makes no sense” Kenneth G. Wilson Finite naturalness

The SM satisfies Finite Naturalness, for the measured values of Mh ≈ Mt.

FN would be ruined by new heavy particles too coupled to the SM. Unlike in the other scenarios, high-scale model building is very constrained. Imagine there is no GUT. No flavour models too. Above us only sky.

Data demand some new physics: DM, neutrino masses, maybe axions...

Can this be added compatibly with FN? Finite Naturalness and new physics

Neutrino mass models add extra particles with mass M  7 √3  0.7 10 GeV√× ∆ type I see-saw model, <  M ∼ 200 GeV × √∆ type II see-saw model,  940 GeV × ∆ type III see-saw model. Leptogenesis is compatible with FN only in type I.

> 9 Axion and LHC usually are like fish and bicycle because fa∼ 10 GeV. Axion models can satisfy FN, e.g. KSVZ models employ heavy quarks with mass M  ¯ √  0.74 TeV if Ψ = Q ⊕ Q <  ¯ M ∼ ∆ × 4.5 TeV if Ψ = U ⊕ U  9.1 TeV if Ψ = D ⊕ D¯ Inflation: flatness implies small couplings. Einstein gravity gives an upper bound on HI and on any mass [Arvinataki, Dimopoulos..]

t

g 2 6 2 H H yt M 1/6 14 δm ∼ M M ∼ so M <∆ × 10 GeV M4 (4π)6 ∼ g Pl

t

Dark Matter: below about a TeV if weakly coupled. DM with EW gauge interactions

2 Consider a Minimal Dark Matter n-plet. 2-loop quantum corrections to Mh :  M2 cnM2 n2 − 1  6 ln − 1 for a fermion 2 = 4 + 2 4 Λ2 δm 4( g2 Y gY ) × 3 2 M2 M2 7 (4π) 4  ln + 2 ln + for a scalar  2 Λµ2 Λ2 2

Quantum numbers DM could DM mass mDM± − mDM Finite naturalness σSI in SU(2) U(1) Spin decay into in TeV in MeV bound in TeV, Λ ∼ M 10−46 cm2 L Y √ Pl −2 2 1/2 0 EL 0.54 350 0.4 × √∆ (2.3 ± 0.3) 10 2 1 2 1 2 1 1 341 1 9 ∆ (2 5 0 8) 10−2 / / EH . . × √ . ± . ∗ 3 0 0 HH 2.5 166 0.22 ×√ ∆ 0.60 ± 0.04 3 0 1/2 LH 2.7 166 1.0 × √∆ 0.60 ± 0.04 3 1 0 HH, LL 1.6+ 540 0.22 ×√ ∆ 0.06 ± 0.02 3 1 1 2 1 9+ 526 1 0 ∆ 0 06 0 02 / LH . . × √ . ± . ∗ 4 1/2 0 HHH 2.4+ 353 0.14 ×√ ∆ 1.7 ± 0.1 ∗ 4 1/2 1/2 (LHH ) 2.4+ 347 0.6 × √∆ 1.7 ± 0.1 4 3/2 0 HHH 2.9+ 729 0.14 ×√ ∆ 0.08 ± 0.04 4 3 2 1 2 ( ) 2 6+ 712 0 6 ∆ 0 08 0 04 / / LHH . . × √ . ± . ∗ ∗ 5 0 0 (HHH H ) 9.4 166 0.10 ×√ ∆ 5.4 ± 0.4 5 0 1 2 stable 10 166 0 4 ∆ 5 4 0 4 / . × √ . ± . 7 0 0 stable 25 166 0.06 × ∆ 22 ± 2 A new principle

Finite Naturalness is phenomenologically viable, what about its theory? Nature has no scale

FN needs something different from the effective field theory ideology 6 4 2 2 H L ∼ Λ + Λ |H| + L + + ··· 4 Λ2 that leads to the hierarchy problem. Nature is singling out L4. Why? Principle: “Nature has no fundamental scales Λ”.

Then, the fundamental QFT is described by L4: only a-dimensional couplings.

We assumed no masses. But power divergences have mass dimension. So they must vanish: R dE E = 0. Anything different is dimensionally wrong.

Quantum corrections break scale invariance and should generate Mh,MPl

Can this happen? I apply this principle first to matter and later to gravity. What is the weak scale?

◦ Could be the only scale of particle physics. Just so.

• Could be generated from nothing by heavier particles.

- See-saw, axions, gravity...

• Could be generated from nothing by weak-scale dynamics.

- Another gauge group might become strong around 1 TeV.

- The quartic of another scalar might run negative around 1 TeV. WEAKLY COUPLED MODELS

The Coleman-Weinberg mechanism can dynamically generate the weak scale

Model [Hambye, Strumia, 1306.2329]: GSM ⊗ SU(2)X with one extra scalar S, doublet under SU(2)X and potential 4 2 4 V = λH|H| − λHS|HS| + λS|S| .

1) Dynamically generates the weak scale and weak scale DM 2) Preserves the successful automatic features of the SM: B, L... 3) Gets DM stability as one extra automatic feature. Weakly coupled SU(2) model

1) λS runs negative at low energy: 1 g 4 X g3 s 9g g2 X g1 λ ' β ln with β ' yt S λS λS 2 s∗ 8(4π) 10-1 ! ΛH 1 0 −1/4 S(x) = √ w ' s∗e 2 w + s(x) ΛS Couplings s 10-2 1 0 ! λ ( ) = √ HS H x v ' w ΛHS 2 v + h(x) 2λH

10-3 102 104 106 108 1010 1012 1014 1016 1018 1020 2) No new Yukawas. RGE scale Μ in GeV

1 3) SU(2)X vectors get mass MX = 2gXw and are automatically stable.

4) Bonus: threshold effect stabilises λ = λ + λ2 /β . H HS λS Experimental implications

1) New scalar s: like another h with suppressed couplings; s → hh if Ms > 2Mh. 2) Dark Matter coupled to s, h. Assuming that DM is a thermal relict 2 2 3 1 11gX gX 26 cm σvann + σvsemi ann = + ≈ 2.2 × 10 2 − 1728πw2 64πw2 s fixes gX = w/2 TeV, so all is predicted in terms of one parameter e.g. gX:

100 1.4 1042 LEP Bounds ATLAS, CMS Bounds

1.3 2 cm 1043 101 in units 1.2 Xenon2012 Higgs

1.1 section  Higgs 10 44 SM 102 SM X cross

1. g in 1045 LUX2013 0.9 3

section 10

0.8 independent  1046 Cross

0.7 Spin 104 47 0.6 10 100 200 300 400 101 102 103 Mass in GeV of the extra Higgs Mass of the DM candidate in GeV Dark/EW phase transition is 1st order: gravitational waves, axiogenesis? STRONGLY COUPLED MODELS

Techni-Color can dynamically generate a mass scale for an elementary Higgs

Model: [Anitipin, Redi, Strumia, 1410.1817] 4 GSM⊗SU(N)TC with one extra fermion in the (0Y , 3L, 1c,N ⊕N¯). V = λH|H|

No extra scalars, no masses: as many parameters as the SM! The weak scale from strong dynamics

The Higgs mass arises as 9g4 Z d4q Π (q2) 9g4 Z m2 = − 2 WW = 2 dQ2Π (−Q2). 4i (2π)4 q2 4(4π)2 WW where strong dynamics is parameterised by ΠWW : Z 4 iq·x a b ab 2 2 i d x e h0|TJµ(x)Jν(0)|0i ≡ δ (q gµν − qµqν)ΠWW (q ).

Weak coupling Strong coupling at high energy Strong coupling XGa2\ Q ΜΝ

Ρ

W W W W W W H H H H H H

Q Q

H H H The weak scale from strong dynamics

• Dispersion relations show that m2 < 0: ∼−σ<0 z }| { ∂Π q2 ∂Π ∂Π (q2) 1 Z ∞ Im Π (s) VV = VV WW = WW 0 2 − 2 2 , 2 ds 2 2 < ∂ΛTC ΛTC ∂q ∂q π 0 (s − q )

• Operator Product Expansions demand that m2 is UV-finite

q2Λ2 2 TC 2 2 αTC A2 Π (q ) ' c1(q ) + c3(q ) h0| G |0i + ··· WW 4π µν | {z } | {z } | {z } dimensionless −C/q4 positive

• Vector Meson Dominance estimates m2 g4m2 Π ( 2) = ρ 2 2 ρ WW q 2 2 2 ⇒ m ∼ − 2 2 gρ (q − mρ + i) (4π) gρ

s g2mρ 3 2 Natural result: mρ ∼ 20 TeV, m ∼ 50 TeV, mπ ≈ (n − 1) ∼ 2 TeV. B n 4π 4 Pions in the 3 ⊗ 3 − 1 = 3 ⊕ 5 of SU(2)L. π5 decays via the anomaly π5 → WW . Dark Matter from strong dynamics

The model has two DM candidates:

• The lightest techni-baryon, pre- -42 sumably subdominant: its thermal 10 abundance reproduces the DM -43 abundance for m ∼ 200 TeV 10 bound B 2 CL 4 2 % techni (estimate: σv ∼ gTC/4πM e.g. cm 90 for

2 in -44 g - σpp¯v ∼ 100/mp). Its magnetic 10 = baryon SI LUX2013 1 dipole can give a detectable signal Σ or g DM -45 E with a characteristic recoil energy 10 = 0.0003 dependence. section 10-46 cross • The techni-pion π3, stable be- * DM techni-pion DM cause of accidental -parity -47 G Q → 10 HMDM tripletL 2 c exp(iπT )Q . It has a negligible Ν background quartic behaving as Minimal Dark 10-48 Matter: mπ3 = 2.5 TeV, σSI ≈ 100 101 102 103 104 105 0 23 10−46 cm2. . Dark Matter mass in GeV SU(N)TC composite Dark Matter

SU(N) techni-color. Yukawa Allowed Techni- Techni- Techni-quarks couplings N pions baryons under NTF = 3 8 8, 6¯,... for N = 3, 4,... SU(3)TF Q = V 0 3 3 VVV = 3 SU(2)L N∗ Q = N ⊕ L 1 3, .., 14 unstable N = 1 SU(2)L 0 NTF = 4 15 20, 20 ,... SU(4)TF Q = V ⊕ N 0 3 3 × 3 VVV,VNN = 3,VVN = 1 SU(2)L N∗ Q = N ⊕ L ⊕ E˜ 2 3, 4, 5 unstable N = 1 SU(2)L NTF = 5 24 40, 50 SU(5)TF Q = V ⊕ L 1 3 unstable VVV = 3 SU(2)L Q = N ⊕ L ⊕ L˜ 2 3 unstable NLL˜ = 1 SU(2)L = 2 4 unstable NNLL,˜ LLL˜ L˜ = 1 SU(2)L 0 NTF = 6 35 70, 105 SU(6)TF Q = V ⊕ L ⊕ N 2 3 unstable VVV,VNN = 3,VVN = 1 SU(2)L Q = V ⊕ L ⊕ E˜ 2 3 unstable VVV = 3 SU(2)L Q = N ⊕ L ⊕ L˜ ⊕ E˜ 3 3 unstable NLL,˜ L˜L˜E˜ = 1 SU(2)L = 3 4 unstable NNLL,˜ LLL˜ L,˜ NE˜L˜L˜ = 1 SU(2)L NTF = 7 48 112 SU(7)TF Q = L ⊕ L˜ ⊕ E ⊕ E˜ ⊕ N 4 3 unstable LLE, L˜L˜E,L˜ LN,˜ EEN˜ = 1 SU(2)L Q = N ⊕ L ⊕ E˜ ⊕ V 3 3 unstable VVV,VNN = 3,VVN = 1 SU(2)L NTF = 9 80 240 SU(9)TF Q = Q ⊕ D˜ 1 3 unstable QQD˜ = 1 SU(2)L NTF = 12 143 572 SU(12)TF Q = Q ⊕ D˜ ⊕ U˜ 2 3 unstable QQD,˜ D˜D˜U˜ = 1 SU(2)L SO(N)TC composite Dark Matter

SO(N) techni-color. Yukawa Allowed Techni- Techni- Techni-quarks couplings N pions baryons under NTF = 3 5 3, 1, ... for N = 3, 4, ... SO(3)TF N Q = V 0 3, 4, .., 7 unstable V = 3, 1, ... SU(2)L NTF = 4 9 4, 1, ... SO(4)TF Q = N ⊕ V 0 3, 4, .., 7 3 VVN = 1,V (VV + NN) = 3, SU(2)L VV (VV + NN) = 1, ... SU(2)L NTF = 5 14 5, 1... SO(5)TF Q = L ⊕ N 1 3, 4, .., 14 unstable LLN¯ = 1, SU(2)L LL¯(LL¯ + NN) = 1, ... SU(2)L NTF = 7 27 1, ... SO(7)TF 2 Q = L ⊕ V 1 4 unstable (LL¯ + VV ) = 1 SU(2)L 2 Q = L ⊕ E ⊕ N 2 4, 5 unstable (EE¯ + LL¯) + NN(LL¯ + EE¯) = 1 SU(2)L NTF = 8 35 1 SO(8)TF Q = G 0 4 unstable GGGG = 1 SU(2)L 2 Q = L ⊕ N ⊕ V 2 4 unstable (LL¯ + VV ) + NN(LL¯ + VV ) = 1 SU(2)L NTF = 9 44 1 SO(9)TF 2 Q = L ⊕ E ⊕ V 2 4 unstable (EE¯ + LL¯ + VV ) = 1 SU(2)L NTF = 10 54 1 SO(10)TF Q = L ⊕ E ⊕ V ⊕ N 3 4 unstable as L ⊕ E ⊕ V + NN(LL¯ + EE¯ + VV ) = 1 SU(2)L   gSMΛUV Usual naturalness Mh & δMh ∼  gextraMextra Finite naturalness

100 Unnatural

Natural

10-5 Higgs the to

Pl M E 10-10 » Coupling

g gravity

10-15 102 104 106 108 1010 1012 1014 1016 1018 1020 Energy in GeV Agravity What about gravity?

2 2 Does quantum gravity give δMh ∼ MPl ruining Finite Naturalness?

−1 Maybe MPl is just a small coupling and there are no new particles around MPl.

Quantum gravity would be very different from what strings suggest...

[Salvio, Strumia, 1403.4226] Adimensional gravity

Applying the adimensional principle to the SM plus gravity and a scalar S gives: Z 4 q S = d x | det g| L

2 2 2 R R − 3Rµν 2 2 4 2 L = LSM + 2 + 2 + |DµS| − ξS|S| R − λS|S| + λHS|HS| 3f0 3f2 where f0, f2 are the adimensional ‘gauge couplings’ of gravity and R ∼ ∂µ∂νgµν.

Of course the theory is renormalizable, and indeed the graviton propagator is:   −i 2 (spin 2) 2 (spin 0) 2f Pµνρσ − f Pµνρσ + gauge-fixing . k4 2 0 2 ¯ 2 The Planck scale should be generated dynamically as ξShSi = MPl/2.

Then, the spin-0 part of g gets a mass M ∼ f M and the spin 2 part splits µν 0 0 Pl √ into the usual graviton and an anti-graviton with mass M2 = f2M¯Pl/ 2 that acts as a Pauli-Villars in view its negative kinetic term [Stelle, 1977]. A ghost?

In presence of masses, ∂4 can be decomposed as 2 fields with 2 derivatives: 1 1 1 " 1 1 # = 4 → 4 2 2 2 2 − 2 2 k k − M2 k M2 k k − M2 Ostrogradski showed in 1850 that higher derivatives are always bad:

∂4 ⇒ unbounded negative energy ⇒ the classical theory is dead.

Who cares, nature is quantum. ∂4 can be quantized as: i) negative energy, or as ii) negative norm and positive energy. This is the i choice that makes agravity renormalizable. This is demanded by path-integral geometry.

A non-sense or just a slightly acausal unitary S matrix?

For the moment, let’s ignore the issue and compute. Anti-particles teach us that sometimes we get the right equations before understanding their meaning. A ghost? Quantum Agravity...

The quantum behaviour of a renormalizable theory is encoded in its RGE. The unusual 1/k4 makes easy to get signs wrong. Literature is contradictory.

• f2 is asymptotically free: 2 " # df 133 N N Ns (4π)2 2 = −f 4 + V + f + d ln µ 2 10 5 20 60 • Gravity does not affect running of gauge couplings: these two diagrams cancel

V g

V V V V

g

presumably because abelian g is undefined without charged particles.

2 • f0 is not asymptotically free unless f0 < 0 2 4 2 df0 5 4 2 2 5 4 f0 X 2 (4π) = f2 + 5f2 f0 + f0 + (1 + 6ξs) d ln µ 3 6 12 s ...Quantum Agravity

• Yukawa couplings get an extra multiplicative RGE correction:

2 dyt 9 3 2 15 2 (4π) = y − yt(8g − f ) d ln µ 2 t 3 8 2

• RGE for ξ 4   2 dξH 5f2 2 2 2 3 2 (4π) = − 2ξH + f0 ξH(6ξH + 1)(ξH + ) + (6ξH + 1) 2yt − g2 + ··· d ln µ 3f0 3 4

• Agravity makes quartics small at low energy: dλ 9 (4π)2 H = ξ2 [5f 4 + f 4(1 + 6ξ )2] − 6y4 + g4 + ··· d ln µ H 2 0 H t 8 2

• Agravity creates a mixed quartic: dλ ξ ξ (4π)2 HS = H S [5f 4 + f 4(6ξ + 1)(6ξ + 1)] + multiplicative d ln µ 2 2 0 S H Generation of MPl

Mechanisms that can generate dynamically the Planck scale:

Non-perturbative: Some coupling g runs non-perturbative at MPl Perturbative: Some quartic λS runs negative at MPl Non-perturbative models are easily built. Add to the SM:

• An extra gauge group GTG that becomes strong at ΛTG ∼ MPl. • No scalars; no fermions charged under both GSM and GTG.

2 The sign of MPl seems to depend on the (uncomputable?) strong dynamics.

4 The cosmological constant tends to be V ∼ −MPl; specific models avoid it (e.g. adding a fermion in the adjoint of GTG). Generation of MPl: perturbative

1 4 Add a scalar Planckion s with quantum potential V (s) ≈ 4λS(¯µ ∼ s)s . Usually it gets a Coleman-Weinberg minimum when λS runs negative. The gravitational coupling ξS makes the vacuum equation non-standard: ∂V 4V ∂V − = 0 i.e. E = 0 ∂s s ∂s 2 2 2 where VE = V/(ξSs ) ∼ λS(s)/ξS(s) is the Einstein-frame potential. The vev q hsi = M¯Pl/ ξS needs a condition different from the usual Coleman-Weinberg:

βλ (¯µ ∼ hSi) βξ (¯µ ∼ hSi) S − 2 S = 0 λS(¯µ ∼ hSi) ξS(¯µ ∼ hSi) The cosmological constant vanishes if

λS(¯µ ∼ hsi) = 0 Then the minimum equation simplifies to

β (¯µ ∼ hsi) = 0 λS

Is this fine-tuned running possible? This is how λH runs in the SM

RGE running of the MS quartic Higgs coupling in the SM

0.03 0.0003 3Σ bands in 0.02 Mt = 173.3 ± 0.8 GeV HgrayL 0.0002 ΑsHMZL = 0.1184 ± 0.0007HredL Mh = 125.1 ± 0.2 GeV HblueL 0.01 0.0001 H H 0.00 Λ 0.0000 Λ Β

-0.01 -0.0001

-0.02 -0.0002

-0.0003 - 0.03 1016 1017 1018 1019 105 1010 1015 1020 1025 1030 RGE scale Μ in GeV RGE scale Μ in GeV

We do not live in the h ∼ 1017.5 GeV minimum. Another scalar needed: a SM mirror, or something else with gauge and Yukawa interactions. Generation of the Weak scale

RGE running between M0,2 and MPl: 2 dM h i (4π)2 h = −ξ 5f 4 + f 4(1 + 6ξ ) M¯ 2 + ··· d lnµ ¯ H 2 0 H Pl

q −8 11 The weak scale arises if f0,2 ∼ Mh/MPl ∼ 10 i.e. M0,2 ∼ 10 GeV 4 All small parameters such as f0,2 and λHS ∼ f0,2 are naturally small The other side of the Quantum Quantisation of 4-derivative systems

Quantisation was first understood for spin 1 and 0 particles with 2 derivatives.

Spin 1/2 fields have 1 derivative. L = Ψ[¯ i∂/−m]Ψ classically leads to negative Z 3 d p † † H = Ep[a ap,s − bp,sb ] (2π)3 p,s p,s Quantisation allows positive energy.

4 derivative systems have the same problem: negative classical energy.

‘Ghosts’ are avoided like a plague by serious theorists and explored only by crackpots such as Dirac, Pauli, Heisenberg, Pais and Uhlenbeck, Lee, Wick and Cutkosky, Coleman, Feynman, Boulware and Gross, Hawking and Hertog...

4 Goal: find quantization of ∂ . A unique scheme emerges, where i~ and T-odd coordinates play the key role [Salvio, Strumia, to appear]. Predictions for inflation Introduction

Consider a generic inflaton s " 2 # q R (∂µs) L = det g −f(s) + − V (s) + ··· 2 2 E ¯ 2 Make gravity canonical via a Weyl transformation gµν = gµν × MPl/f:    ¯ 2 02! 2  q  MPl 2 1 3f (∂µs)  L = det g − R + M¯ + −V + ··· E  2 E Pl 2 2 2 E   f f  | {z } If desired make s canonical ¯ 4 2 where VE = MPlV/f is the Einstein-frame potential. If V and f are generic functions, VE is generic: ad hoc assumptions were invoked to make VE flat.

2 1 4 In quantum agravity f = ξS(¯µ ∼ s)s and V = 4λS(¯µ ∼ s)s 1 ¯ 4 2 So VE = 4MPlλS(s)/ξS(s) is quasi-flat, even above MPl.

. Inflation = perturbative agravity

Inflation is not a generic phenomenon: one needs to flatten potentials or justify hilltop initial conditions or consider super-Planckian field variations, which are forbidden in string theory where the scalar field space is compact with MPl size.

Inflation becomes a generic phenomenon in agravity: the slow-roll parameters are given by the β-functions, which are small if the theory is perturbative. E.g. " #2 1 ξ βλ βξ  = S S − 2 S , 21 + 6ξS λS ξS Inflaton candidates in agravity

In agravity all scalars can be inflatons, and there are at least 3 scalars:

s The scalar ‘Planckion’ that breaks scale invariance generating MPl. It can be light, being the pseudo-Goldstone boson of scale invariance:

2 2 Ms ∼ gs MPl/(4π)

If it is the inflation one has ns ≈ 0.967 and r ≈ 0.13.

z The scalar component of the graviton, M0 ∼ f0MPl. If it is the inflaton one has Starobinski inflation: ns ≈ 0.967 and r ≈ 0.003.

h The Higgs. If it is the inflation one has Higgs inflation: ns ≈ 0.967 and r ≈ 0.003? Limit I: Planckion inflation

The Planckion could be so light that agravity can be integrated out leaving s4 s2 L = −λ − ξ R + gauge and Yukawas eff S 4 S 2 Running that generates M¯Pl with Λ = 0 (fermion in the inflaton sector needed): g4 s λ (¯µ ≈ s) ≈ ln2 + ··· , ξ (¯µ) ≈ ξ + ··· S 2(4π)4 hsi S S

The canonical Einstein-frame field is s 1 + 6ξS s sE = M¯Pl ln ξS hsi Its potential is quadratic: ¯ 4 2 2 ¯ MPl λS Ms 2 g MPl 1 V = ≈ s with Ms = E 4 ξ2 2 E 2(4π)2 q S ξS(1 + 6ξS) Limit I: Planckion inflation √ Inflation occurs at sE ≈ ±2 NM¯Pl for N ≈ 60: above the Planck scale: g4N2 2 A 8 1 0 967 = t 0 13 As ≈ 2 ns ≈ − ≈ . , r ≈ ≈ . . 24π ξS(1 + 6ξS) N As N

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�������� ������ � 4 17 In the SM-mirror model b ≈ 1.0/(4π) so ξS ≈ 230 so hsi ≈ 1.6 10 GeV: ok. Limit II: graviton inflation

If M0  Ms the Planckion is frozen at its minimum " ¯ 2 2 # q MPl R Leff = det g − R + 2 + LWeyl 2 6f0 The Weyl term does not contribute to FRW. One gets Starobinski inflation.

E 2 ¯ 2 After a Weyl rescaling gµν = gµν × z /6MPl  ¯ 2 2 ¯ 2 !2 q MPl 2 (∂µz) 4 3 2 MPl Leff = det g − R + 6M¯ − M¯ f 1 − 6  E 2 E Pl 2z2 Pl8 0 z2

Ht Pure agravity, M¯Pl = 0, gives a ∝ e with arbitrary H. Generic case: M0 ∼ Ms

2 2 2 The full computation is done converting ξSs R and R /f0 to the Einstein frame.

2 2 2 2 2 Add an auxiliary field χ and 0 = −(R + 3f0 χ/2) /6f0 to cancel R /f0 , get  1 2 2 2  q 3R − Rµν f 3f0 2 2 2 2 2 L = | det g| Lmatter + 2 − R − χ  f = χ + ξHh + ξSs . f2 2 8 Make gravity canonical with a Weyl rescaling gE = g × f/M¯ 2 . The scalar µν µν Pl √ ¯ 2 kinetic metric has constant negative curvature ∼ −MPl. Define z = 6f: 2 2 2 2 6M¯ (Dµh) + (Dµs) + (∂µz) L 3 Pl . z2 2 The potential is non-trivial:   ¯ 4 2 2 !2 36MPl 3f0 z 2 2 VE(z, s, h) = V (h, s) + − ξHh − ξSs  . z4 | {z } 8 6  quartic

• For large f0, the second term vanishes, giving Planckion inflation. • For negligible f0, s stays fixed and z gives Starobinski inflation. Who is the inflaton?

Predictions might depend on the initial condition. We find that, whatever is the starting point, slow-roll converges towards a unique attractor solution, probably because a dimensionless-potential has V 00 ∼ λ field2.

ΞS = 1 , MsM0 = 1.00

Ξ = Ξ =  = Λ = S 1, H 1 , Ms M0 0.10, H 0.01 N=80 N=60 30 N=40 N=20 Pl M

15 

z 10 field

10 3 Higgs graviton hM Pl 1 5 Scalar 5

zM Pl 10 0.3 0 NN=N=80=N6040=20 15 3 2 1 5 4 0.3 1 3 Planckion sM Pl Planckion field sM Pl

The Higgs is never relevant because of its large λH. Predictions for inflation

Predictions of agravity inflation Predictions of agravity inflation 0.3 0.3 N = 50 ΞS = 80.1,1,10< 68,95% C.L. ΞS = 0.1 Hdot-dashedL ΞS = 1 HcontinuousL 0.1 ΞS = 10 HdottedL N = 60 0.1 r r ΞS = 100 HdashedL ratio ratio 0.03 0.03 scalar scalar  

Tensor 0.01 0.01 Tensor

N = 50 0.003 0.003 N = 60

10-2 10-1 100 101 102 0.92 0.94 0.96 0.98 1.00

M0Ms Spectral index ns [Kannike, Hutsi, Pizza, Racioppi, Raidal, Salvio, Strumia, 1502.01334] Decay of the inflaton: Einstein frame

The Einstein-frame can be confusing: consider a Yukawa of s to a fermion ψ:

q " s4 # det g λ + y sψψ + ψi¯ Dψ/ + ··· S 4 Going to the Einstein frame with a Weyl transformation

E −1/2 −3/4 gµν = gµν × Ω, sE = s × Ω , ψE = ψ × Ω ,AµE = Aµ. 2 ¯ 2 with Ω = ξSs /MPl one gets " # q s4 det g λ E + ys ψ ψ + ψ¯ iD/ ψ + ··· E S 4 E E E E E E √ Notice: sE = M¯Pl/ ξS = cte, all non-gravitational couplings of s disappear???

Somehow, these become gravitational couplings of ψE with mass ∼ ysE, etc.

Better to understand in which sense a dimension-less theory is special... Decay of the inflaton: Jordan frame

The inflaton is a combination of s and z. They decay in a similar way

∂µDµ Tµµ √ and M¯Pl/ ξS M¯Pl Since in the Jordan frame the theory is explicitly dimension-less, the divergence 0 of Dµ = Tµνxν + Dµ vanishes at tree level (after using manipulations analogous to those used for a global U(1) phase symmetry e.g. 2m ΨΨ¯ = ∂µ[Ψ¯ γµγ5Ψ]). At loop level scale invariance is broken by quantum corrections:

βg βg βg ∂ = 1 Y 2 + 2 W 2 + 3 G2 + β HQ U + β |H|4 + ··· µDµ µν µν µν yt 3 3 λH 2g1 2g2 2g3

Since the theory is not conformal, Tµµ has a tree level term: s 2 2 ξS hE ∂ sE −(1 + 6ξH) 1 + 6ξS 2 M¯Pl 2 3 ¯ 2 7−9 Final result Γ = (tree or loop) × Ms /MPl and TRH = 10 GeV Dark Matter and Leptogenesis

Furthermore, the inflaton could have sizeable or dominant decays to possibly lighter particles within its sector, that must contain a fermion ψ.

From the SM point of view, ψ is a ‘right-handed neutrino’. So ψ could naturally couple via ψLH and decay giving leptogenesis.

If ψLH is not allowed (e.g. if ψ has dark gauge interactions), then ψ is is stable. ¯ 2 ¯ 2 It behaves as DM. If Mψ breaks scale symmetry, then Γ(s → ψψ) ≈ MψMs/MPl, such that the DM abundance is reproduced for 2/3  Ms  M ≈ (10 − 200) TeV . ψ 1013 GeV ¯ PR ∼ Mh/MPl

−9 Any superPlanckian theory gives inflation, also eternal: but PR ∼ 10  1?

Agravity relates the smallness of the amplitude of inflationary perturbations PR ¯ to the smallness of Mh/MPl, up to couplings and loops and powers of N ≈ 60.

Consider ainflation in the Starobinski limit: 2 2 f0 N −5 P = i.e. f0 = 1.8 10 . R 48π2 The quantum correction to the Higgs mass is dominated by the RGE: dM2 h = −ξ (1 + 6ξ )f 4M¯ 2 + ··· d lnµ ¯ H H 0 Pl < −5−8 So finite naturalness demands f0∼ 10 (at tree-loop level).

1 In minimal models, the two values of f0 are compatible if ξ is close to 0 or −6. Towards infinity Landau poles in soft gravity

2 2 If the theory has no cut-off Λ, it cannot give δMh ∼ Λ We have the RGE above MPl, can the theory reach infinite energy? Obstacle: Landau poles 43 In the SM gY → ∞ at 10 GeV. λ, yt, yb, yτ can hit other poles. 2 2 We assume that Landau poles give δMh ∼ MLandau and must be avoided.

We view agravity as an example of the more generic scenario of soft-gravity: < 11 the gravitational coupling g ∼ E/MPl stops growing at E∼ 10 GeV when it is 2 still small enough that gravitational corrections to Mh are naturally small. The RGE of soft-gravity are dominated by the bigger SM couplings like in agravity.

[Giudice, Isidori, Salvio, Strumia, 1412.2769] Total Asymptotic Freedom?

Goal: compute if/when all n  1 couplings of a realistic QFT can run up to E = ∞.

Result: too hot for eggs Naive attempt:

• solve the RGE for g, y, λ numerically

• up to infinite energy

• identify m-dimensional sub-spaces.

Analytic tools needed TAFfing procedure

2 2 ˜ Rewrite RGE in terms of t = ln µ /(4π) and of xI = {˜gi, y˜a, λm} as 2 2 ˜ 2 ˜gi (t) 2 y˜a (t) λm(t) g (t) = , y (t) = , λm(t) = . i t a t t Get   ˜gi/2 + βgi(˜g), dxI  = VI(x) = y˜a/2 + βya(˜g, y˜), d ln t  ˜ ˜ λm + βλm(˜g, y,˜ λ). Fixed-points xI(t) = x∞ are determined by the algebraic equation VI(x∞) = 0.

Linearize around each fixed-point:

∂V V (x) ' X M (x − x ) where M = I I IJ J J∞ IJ ∂x J J x=x∞ Negative eigenvalues of M are UV-attractive. Each positive eigenvalue implies a UV-repulsive direction: to reach the FP a coupling is univocally predicted. Flows in the SM with gY = 0

0.25 0.25

0.20 0.20 IR

0.15 IR 0.15 2 t 2 t y y t t

0.10 0.10

0.05 0.05

0.00 UV 0.00

0.00 0.05 0.10 0.15 0.20 0.25 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 2 t yb t Λ

2 2 2 2 ˜λ∞ Eig y˜t∞ y˜b∞ y˜τ∞ y˜ν∞ Eigenvalues √ Solution 1 227/1197 0 0 0 +−++ −143 + 119402 Sol. 1 ≈ +0.0423 + Solution 2 0 227/1197 0 0 −+++ 4788√ Solution 3 227/1596 227/1596 0 0 ++++ −143 − 119402 Solution 4 0 0 0 0 −−++ Sol. 2 ≈ −0.1020 − 4788 SM up to infinite energy?

2 Predictions: 1) gY = 0; in√ this limit 2) yt ' 227/1197t i.e. Mt = 186 GeV; 3) yτ,ν = 0; 4) λ ' (−143 ± 119402)/4788t i.e. Mh ≤ 163 GeV. Equality avoids λ < 0 at large energy, and too fast vacuum decay λ < −1/12t.

SM for g1 = 0 and Mt = 185.6 GeV

1 g3

g2 yt

0.3

ÈΛÈ 0.1 couplings

SM 0.03

y 0.01 Τ yb

0.003 101 1010 10100 101000 1010000 10100000 RGE scale Μ in GeV TAF extensions of the SM Can the SM be extended into a theory valid up to infinite energy? Avoid Landau poles by making hypercharge non abelian.

We found realistic SU(5) TAF models. But GUTs are not compatible with finite naturalness, that demands a TAF extension at the weak scale. Making sense of Y = T3R + (B − L)/2 needs SU(2)R. We see 2 possibilities:

SU(4)c ⊗ SU(2)L ⊗ SU(2)R and SU(3)c ⊗ SU(3)L ⊗ SU(3)R

SUH2LL SUH3LL SUH2L R SUH3LR eL ΝL e'L Ν' Ν'L Ν L  ΝR eL ΝR eR e ΝL R  dR eL uR d R  dR d uL SUH3L uL R SUH4LPS c uR uR dL dL  dR  dR dR dR uL uL uR d uR L dR dL  uL dR dR uL u dL R dL  dR TAF is tough

Too many experimental signals

0 • A WR boson and a Z : MW > 2.2 TeV, M 0 > 2.6333, 3.8224 TeV B−L R ∼ ZB−L 2 2 2 !2 9gRMW MW MW δM2 = − R ln( R ) ≈ M2 R h (4π)2 µ¯2 h 2.5 TeV

• The Higgs (2L, 2¯R) contains 2 doublets coupled to u and d: new flavour violations controlled by a right-handed CKM matrix. ( 18 TeV if VR = VCKM MH > ij ij 3 TeV if VR = VCKM × min(mi, mj)/ max(mi, mj) (natural texture)

• A lighter singlet that mixes with the higgs if GTAF → GSM dynamically.

• And we still have to find models where all Yukawas and quartics obey TAF Pati-Salam

SUH2LL

SUH2LR

ΝR

e ΝL R

eL uR

dR uL SUH4LPS uR dL

dR uL uR dL

dR uL

dL Pati-Salam

Fields spin generations SU(2)L SU(2)R SU(4)PS   νL eL ψL = 1/2 3 2¯ 1 4 uL dL   νR uR ψR = 1/2 3 1 2 4¯ eR dR φR 0 1 1 2 4¯  0 +  HU HD ¯ φ = − 0 0 2 2 2 1 HU HD ψ 1/2 1, 2, 3 2 2¯ 1 QL 1/2 2 1 1 10 QR 1/2 2 1 1 10 Σ 0 1 1 1 15

No extra chiral fermions. Two ways to get acceptable fermions masses:

c 1) Foot: add ψ and φL: −LY = YN ψLψφR +YL ψψRφL +YU ψRψLφ+YD ψRψLφ . Avoids `L/dL unification so MW 0 > 8.8 TeV. No TAF found for the 24 quartics.

2) Volkas: add QL,R getting dR mixing. Strong flavor bounds MW 0 > 100 TeV because of `L/dL unification. TAF found adding Σ. Trinification

SUH3LL

SUH3LR eL ΝL e'L Ν' Ν'L Ν L  ΝR eL eR

 dR

dR d SUH3L uL R c u dL R  dR  dR dR uL u dL R d  R uL dR u dL R  dR Trinification

Minimal weak-scale trinification model Matter fields gen.s spin SU(3)L SU(3)R SU(3)c 1 2 3 ! uR uR uR 1 2 3 ¯ QR = dR dR dR 3 1/2 1 3 3 d01 d02 d03 R1 R1 ¯0R1 ! uL dL dR 2 2 ¯01 ¯ QL = uL dL dR 3 1/2 3 1 3 u3 d3 d¯03 L0 L0 R ! ν¯L eL eL 0 0 ¯ L = ¯eL νL νL 3 1/2 3 3 1 0 eR νR ν vu 0 0 ! hHi = 0 vd vL 3 0 3 3¯ 1 0 VR V

q 2 2 • Explains quantisation of Y . Needs gR = 2g2gY / 3g2 − gY ≈ 0.65g2. • No bad vectors: V ≈ few TeV allowed. • Extra d0, e0, ν0 fermions chiral under SU(3)3 get mass ∼ yV from Yukawas 1 n ∗ 0 0 0 yQ QLQRH + 2yLLLH . 3H are needed to make d , e , ν naturally heavy. • Di-boson anomalies can be fitted by M ≈ 1.9 TeV with g ∼ 0.6g . WR R L Trinification predicts MZ0 ∼ 3.2 TeV. Trinification TAF

TAF solutions found for H1,H2 (20 quartics) and for H1,H2,H3 (90 quartics).

Extra fermions Extra Yukawas TAF? ∗ 8L 8LLH yes (3gen) ∗ 8R 8RLH yes (3gen) ∗ ∗ 8L ⊕ 8R 8LLH + 8RLH yes (3gen) L0 ⊕ L¯0 L0LH L0L0H + L¯0L¯0H∗ yes (3gen) 0 ¯0 0 QL ⊕ QL QLQRH no 0 ¯0 0 QR ⊕ QR QRQLH no 0 ¯0 0 ∗ QL ⊕ QL ⊕ 8R QLQRH + 8RLH yes 0 ¯0 0 ∗ QR ⊕ QR ⊕ 8L QRQLH + 8LLH yes 0 0 ¯0 ¯0 0 0 0 0 ¯0 ¯0 ∗ QL ⊕ QR ⊕ QL ⊕ QR QLQRH + QRQLHQLQRH + QLQRH yes

[Pelaggi, Strumia, Vignali, 1507.06848]. Trinification Conclusions

Not today.

LHC run II started.

In 3 years we will known more.