Alternatives to Standard Tev-Scale New Physics Scenarios

Alternatives to standard TeV-scale New Physics scenarios

Stéphane Lavignac (IPhT Saclay)

• introduction • usual prejudices about New Physics • split supersymmetry and high-scale supersymmetry • minimal scenarios with sterile neutrinos

« Le futur de la Physique des Particules » Journée de la Division Particules et Champs de la SFP LPNHE, Paris, 23 janvier 2015 Introduction

New Physics expected because the Standard Model is incomplete: Unexplained observational facts: • neutrino masses • dark matter (DM) • matter-antimatter asymmetry (BAU) • inﬂation somewhat different in nature • cosmological constant } involve also gravity Theoretical problems: • strong CP problem • origin of electroweak symmetry breaking (EWSB) • large number of parameters, mass hierarchies • incomplete uniﬁcation of forces • hierarchy problem (in the presence of ⇤ NP M W ) The hierarchy problem has often been used as a guide to discriminate among theories beyond the SM, favouring supersymmetry, extra dimensions and composite Higgs models

These theories are now strongly challenged by negative search results from colliders and dark matter experiments → emergence of alternative scenarios in which naturalness is no longer required, such as: • split supersymmetry • high-scale supersymmetry • scenarios with superheavy sterile neutrinos (seesaw) • scenarios with keV/GeV sterile neutrinos, such as the νMSM 18 C. Grojean be. The SM particles give unnaturally large corrections to the Higgs mass: they destabilize the Higgs vev and tend to push18 it towards the UV cutoff ofC. the Grojean SM. Some precise adjustment (fine-tuning)18 between the bare massC. Grojeanand the one-loop correction is needed to maintain the vevbe.of the The Higgs SM particles around give the unnaturally weak scale: large take corrections to the Higgs mass: they be. The SM particles give unnaturally large corrections to the Higgs mass: they two large numbers, naturally their sum/differencedestabilize the will Higgs be ovevfthesameorderun-and tend to push it towards the UV cutoff of the SM. destabilizeSome the Higgs precisevevUsual adjustmentand tend to (fine-tuning) pushprejudices it towards between the UV the cutoffabout bare of mass the SM.Newand the one-loop Physics less these numbers are almost equal up to several significant digits (see [17] for Some precisecorrection adjustment is needed (fine-tuning) to maintain between the vev the bareof the mass Higgsand around the one-loop the weak scale: take arecentestimateoftheamountoffine-tuningwithintheSMancorrectiontwo is needed large numbers,to maintain naturally the vevdvariousmodelsof their the Higgs sum/difference around the weak will be scale: ofthesameorderun- take BSM). two largeless numbers, these naturally numbers their are almost sum/difference equal up will to be several ofthesameorderun- significant digits (see [17] for less1) these should numbers arebe almost natural equal up ⇒ to several avoid significant the digitshierarchy (see [17] for problem arecentestimateoftheamountoffine-tuningwithintheSMandvariousmodels arecentestimateoftheamountoffine-tuningwithintheSMandvariousmodels BSM). BSM).Origin of the problem: radiative corrections to the scalar mass parameter

= 2 c 2 mH 2 M = = ⇠ 16⇡

Fig. 6. One loop corrections to the Higgs mass. The three diagrams are quadratically divergent and make the Higgs mass highly UV sensitive. Fig. 6. OneFig. loop 6. corrections One loop to corrections the Higgs mass. to the The Higgs three mass. diagrams The are three quadratically diagrams aredivergent quadratically and divergent and makeAny the Higgs heavy mass highly particle UV sensitive. with mass M M W coupling to the Higgs boson will destabilizemake the Higgs the mass weak highly UV scale sensitive. The hierarchy problem is a generic technical problem in any theory involving The hierarchyThe problemhierarchy is problem a generic is technical a generic problem technical in any problem theory involving in any theory involving some light scalar fields. some→ lightthe scalar hierarchy fields. M W M must be ensured by a fine-tuning some light scalar fields. In the study of any theory beyondIn the the study Standard of any theory Model, beyond one need the⌧ Standardstobeableto Model, one needstobeableto In the study of any theory beyond the Standard Model, one needstobeableto quickly estimate the quadraticallyquickly divergent estimate corrections the quadraticallyto the divergent scalar corrections potentials.to the scalar potentials. This quicklycriterion estimate favours the quadratically theories divergent like corrections supersymmetry,to the scalar potentials. extra-dimensions (ADD One can calculate explicitly someOne Feynman can calculate diagrams explicitly or some more Feynman conveniently diagrams rely or more conveniently rely on the computationOne can calculate of the Coleman–Weinberg explicitly some Feynman potential [18].diagramsAt one-loop or more this conveniently rely k⇡rc on the computation of the Coleman–Weinbergmodel where potential M [18]. At1 TeV one-loop , RS this model in which M W /M Pl = e ) and effective potentialon the computation for a scalar( fieldd of) the⇠φ is Coleman–Weinberg given by potential [18]. At one-loop this effective potential for a scalar fieldcompositeφ iseffective given by potentialHiggs formodels a scalar field (solveφ is given the by little hierarchy problem) 4 d kE 2 2 4 V (φ)= 4 STr ln(k4 E +M (φ)), (2.44) d kE 2 2 2(2π) d kE 2 2 V (φ)= STr ln(k +M!(φV))(φ,)= (2.44)STr ln(kE +M (φ)), (2.44) 2(2π)4 E 2(2π)4 ! where the supertrace, i.e. the trace with! an extra minus sign for the fermionic degrees ofwhere freedom, the is supertrace, over all thei.e. particlesthe trace that acquire with an a extra mass whenminusφ signis away for the fermionic where the supertrace, i.e. the trace with an extra minus sign for4 the fermionic from the origin.degrees After of freedom, integrating is over overd allkE the,weget particles that acquire a mass when φ is away degrees of freedom, is over all the particlesfrom thethat origin. acquire After a m integratingass when overφ isd away4k ,weget 4 E from the origin. After integrating over d kE4 ,weget 2 2 Λ Λ 2 1 4 M (φ) V = 2 STr 1 + 2 STr M (φ)+ 2 STr M (φ) ln 2 , −128π 4 64π 2 64π Λ 2 Λ Λ 2 1 4 M (φ) 4 2 V = 2 STr 1 + 2 STr2 M (φ)+ 2 STr M (φ) ln 2 , Λ Λ 2 −1128π 4 64πM (φ) 64π Λ V = 2 STr 1 + 2whereSTr M we( easilyφ)+ read off2 STr the quadratically M (φ) ln divergent2 , corrections to the scalar po- −128π 64π tential. Let us look64 explicitlyπ at the case of theΛ Higgs in the SM. The only things where we easily read off the quadratically divergent corrections to the scalar po- where we easily read off the quadraticallytential. divergent Let us look correc explicitlytions to at the the scalar case of po- the Higgs in the SM. The only things tential. Let us look explicitly at the case of the Higgs in the SM. The only things 2) the dark matter particle is a Weakly Interacting Massive Particle Relic abundance determined by freeze-out:

⌦ 1/ Av / h i Turner] [Kolb, weakly interacting: ↵2 /m2 A ⇠ w ⌦ 0.1 for m (0.1 1) TeV ) ⇠ ⇠ i.e. close to the scale suggested by the hierarchy problem “ WIMP miracle ” Examples: neutralino (if LSP) in supersymmetric models, lightest Kaluza-Klein particle (LKP) in universal extra dimension scenarios, ...

3) neutrino masses are generated from the seesaw mechanism Easily accommodated in any scenario: just add RH neutrinos with a large Majorana mass term and a Dirac coupling to the SM neutrinos ⇒ naturally generates small SM neutrino masses 4) strong CP problem ignored, or solved by the axion ✓ The QCD Lagrangian a priori contains a CP-odd term G G˜µ⌫ 16⇡2 µ⌫ which induces a (large) neutron EDM exp 25 10 d < 0.29 10 ecm ✓ 10 n ⇥ ) . A natural solution involves the pseudo-Goldstone boson of a spontaneously broken, anomalous global symmetry such as the Peccei-Quinn symmetry (the axion a), which drives dynamically θ to zero The axion gets a sub-eV mass from QCD instantons and can constitute part or all of the CDM density

5) the cosmological constant problem is solved somehow If dark energy interpreted as a cosmological constant, observations require 48 4 120 4 ⇤ = V 10 GeV 10 M 0 ⇡ ⇡ Pl → largest fine-tuning in nature. Even if solved, any phase transition (Susy, EW, 4 QCD) occurring at T = ⇤ PT will contribute as ⇤ PT to the vacuum energy Not a problem for the SM alone, but for SM + gravity Among the various possibilities for BSM physics, (low-energy) supersymmetry emerged as the preferred option because: • automatically solves the hierarchy problem • a neutralino LSP is a suitable DM candidate • allows gauge coupling unification • radiative electroweak symmetry breaking possible • cosmological constant problem softened • may be inherited from string theory Extra dimensions also address the hierarchy problem and provide new DM candidates as well as new ideas for EWSB, the flavour problem or the cosmological constant problem Composite Higgs models solve the little hierarchy problem and feature a dynamical origin for EWSB, provide also new ideas for the flavour problem and DM These theories are now strongly challenged by negative search results from the LHC and from dark matter experiments Supersymmetry gluino, first 2 gen. squarks: m> (1 1 .5) TeV (depending on model) Higgs mass constraint: 3g2m4 sin2 m2 X X2 m2 m2 cos2 2 + t ln S + t 1 t h  Z 8⇡2m2 m2 m2 12m2 W  ✓ t ◆ S ✓ s ◆ ⇒ m h = 125 GeV requires multi-TeV stops or maximal stop mixing These lower bounds imply large corrections to the Higgs potential, hence a strong (sub-percent) fine-tuning: 3y2 ⇤2 m2 t m2 ln Hu ⇠ 16⇡2 S m2 ✓ S ◆ Extra dimensions

RS gravitons: m & 2 TeV Large ED (ADD): M(d) & several TeV Composite Higgs models top partners: m (500 800) GeV & (heavier top partners imply a stronger ﬁne-tuning) / Nuclear Physics B Proceedings Supplement 00 (2014) 1–6 4

miss -1 CMS Preliminary e + ET L dt = 20 fb s = 8 TeV 7 ∫ ity between the collider and direct searches. The ﬁrst 10 M = 300 GeV = 200 GeV χ Λ W-> l ν QCD striking feature is the strength of the collider analyses 106 Spin Independent DM ξ = +1 tt + single top γ + jets searching for low-mass dark-matter particles. Indeed, 105 DM ξ = 0 DY Diboson where the recoil signals in the direct searches become 4 10 too soft at low mass for ecient detection, the col- DM ξ = -1 data syst uncer. 103 lider setting allows to maximize the generated missing 102 momentum, and hence sensitivity, at zero mass. An- Events / 1 GeV 10 other complementarity can be seen when comparing the 1 two plots: the direct-detection experiments have typi- 10-1 cally reduced or no sensitivity to spin-dependent interactions, which allows the collider searches to provide 10-2 -3 complemetary coverage also at intermediate masses. At 10 higher mass, the collider searches run out of steam be- -4 10 cause the production cross section drops – here the indi- 500 1000 1500 2000 2500 Darkrect searches matter with neutrino telescopes are probing com- MT (GeV) plementary ground. / Nuclear Physics B Proceedings Supplement 00 (2014) 1–6 5

-1 -36 19.7 fb (8 TeV)

-35 ]

] 10

10 2 2 Figure 3: Distribution of transverse mass after the electron selection CMS Mχ = 50 GeV, Γ = M/3 -36 CMS Preliminary 10 Mχ = 50 GeV, Γ = M/10 described in the text, showing data, estimated background contribu- 10-37 4000 CMS Monolepton ξ = +1 [GeV] Mχ = 50 GeV, Γ = M/8π tions, and signal expectation for an example model of dark matter 10-37 q g Mχ = 500 GeV, Γ = M/3 χ CMS Monophoton, s = 7 TeV, 5.1 fb-1 -38 SIMPLE 2012 production in the three considered cases of interference. -38 g -1 10 ξ=+1 Mχ = 500 GeV, Γ = M/10

10 2014 ICHEP CMS Monophoton, s = 8 TeV, 19.6 fb CMS Monolepton CMS Monojet M = 500 GeV, = M/8 -39 COUPP 2012 χ Γ π - 3000 miss -1 10 + g g contours CMS Preliminary µ + E L dt = 20 fb s = 8 TeV CDMSlite -39 W χ q T ∫ 10 -1 Spin Independent, Vector -40 Super-K W CoGeNT 2011 s = 7 TeV, 5.1 fb 6 M = 300 GeV Λ = 200 GeV 10 -1 µ χ W → l ν tt +single top CMS Monophoton, (χγ χ)(qγ q) 10 = 8 TeV, 19.6 fb µ Spin Independent SuperCDMS s - -41 -40 + W 5 10 SIMPLE 2012 10 CMS Monophoton, 2 10 DM ξ = +1 IceCube W Λ DY QCD 2000 4 10-42 DM ξ = -1 -41 CMS Monojet 10 COUPP 2012 10 Diboson data -43 103 DM ξ = 0 10 CDMS II -44 -42

2 syst uncer. 10 10 M/ on limit CL 90% 10 XENON100 1000 -45 0.1 10 LUX CMS Preliminary -Nucleon Cross Section [cm Events / 1 GeV 10 -Nucleon Cross Section [cm -43 µ 10 µ 0.2 10 χ -46 (χγ χ)(qγ q) χ (χγ γ χ)(qγ γ q) 0.5 1 2 5 10 Spin Independent, Vector Operator µ µ 5 5 1 2 Spin Dependent, Axial-vector operator 2 -47 Λ Λ -1 10 10-44 0 10 2 3 3 2 3 4 1 10 10 10 1 10 102 10 10 10 10 -2 10 Mχ [GeV] Mχ [GeV] Mediator Mass M [GeV] 10-3 10-4 10-5 strongerFigure 5: 90%CL upperand limits stronger on the dark-matter–nucleon constraints scatter- Figure on 6: 90%CL WIMPs upper limits onfrom the dark-matter–nucleon direct detection scattering Figure 7: Limits at 90% CL on the interaction scale ⇤, as a function ing cross section, from the monojet, monophoton, and monolepton cross section, from the monojet, monophoton, and monolepton (⇠ = of the mediator mass in a simpliﬁed model with an s-channel vector 500 1000 1500 2000 2500 experiments(⇠ =+1) searches, as a function and of thesearches dark matter mass, at for spin-the+1) LHC searches, as a function of the dark matter mass, for spin-dependent mediator providing the coupling between the quarrks and the dark- independent (vector operator) interactions MT (GeV) (axial-vector operator) interactions matter particles. Several mass and width assumptions are considered.

The monolepton result shown in Figures 5 and 6 is Figure 4: Distribution of transverse mass after the muon selection de- the most pessimistic case of destructive interference. As tion scale ⇤ is calculated as a function of the mediator miss ET > 350 GeV, and 3 jets, of which one is identi- scribed in the text, showing data, estimated background contributions, can be seen from Figures 3 and 4, the cross section may mass, and a range of decay width for the mediator is fied as a b quark. Additionally, an electron and muon and signal expectation for an example model of dark matter produc- be much higher for other interference scenarios. Corre- considered. In Figure 7, the result of this mediator mass tion in the three considered cases of interference. veto is applied to suppress backgrounnds with genuine spondingly, the limits on the dark-matter–nucleon scat- scan is shown. Three regimes can be discerned. At high missing energy from the neutrino in leptonic W decays. tering cross section from the monolepton search may be mass, the obtained limit coincides with the EFT expec- This selection leaves t¯t and Z+jets as the main back- mass, for spin-independent (vector operator) and spin- much stronger, even surpassing the monojet sensitivity. tation. When decreasing the mediator mass, the media- grounds. The total background expectation is 28 16 ± dependent (axial-vector operator) interactions, respec- More details may be found in [15]. tor can go on-shell, and resonant production boosts the events, while 30 events are observed in the data. In ab- tively. Comparisons are made with results from sev- A first e↵ort has been pursued, in the context of the cross section and hence limit beyond what is naively ex- sence of an excess, limits were determined on the possi- eral direct and indirect detection experiments. While it monojet analysis, to move beyond the EFT interpreta- pected from the EFT approach. For even lower media- ble presence of a scalar and vector dark-matter particle should be stressed to keep the aforementioned caveats tion, and make the mediator explicit by means of a sim- tor masses, the mediator goes o↵-shell again, and the in this monotop scenario. In Figure 8, the 95% CL upper on the interpretation of the EFT resuls in mind, a few plified model. In the studied case, the mediator is con- limit on ⇤ decreases below the naive EFT approxima- limit on the cross section is shown as a function of the robust observations can be made on the complementar- sidered to be a vector particle. The limit on the interaction, making the EFT limit too aggressive with respect mass of the dark-matter candidate, in the case it is a vec- to a realistic model with an explicit mediator. tor particle. This is compared with the production cross section of the considered model, leading to this scenario 5. Searches in final states with top quarks being excluded at 95% CL for masses below 650 GeV. A scalar dark-matter candidate is similarly excluded for Two other searches are presented for dark matter, this masses below 330 GeV. ¯ miss time leading to final states with missing energy and a The selection for the tt + ET final state aims for the single [16] or two top quarks [17]. In the case of a single dilepton decay channel. Two well-identified electrons top quark, referred to as a monotop final state, the dark or muons are required, along with two or more jets, and miss matter particle is assumed long-lived, and couples to ET > 320 GeV. Further cuts are applied on the open- the top quark through flavour-changing diagrams. The ing angle between the leptons, and on the scalar sums of second analysis, looking for two top quarks with miss- the transverse momenta of the leptons on the one hand, ing energy, considers an EFT scenario where the dark- and the jets on the other. The background remaining af- matter preferentially couples to heavy quarks, like is the ter these selection cuts is dominated by top quarks, with case for a scalar interaction with a coupling proportional a non-negligible contribution from diboson and Drell- to the mass of the interacting quark. Yan events. The total background is estimated to be The selection for the monotop search selects hadronic 1.9 0.7 events, while in data 1 event is observed to pass ± final states by requiring a large missing momentum, the selection. With background expectation and data be- Thus we may want to abandon some of our theoretical prejudices

→ Strategy: insist on explaining what we see • DM (not necessarily a WIMP, could be an axion or the gravitino) • the baryon asymmetry of the Universe • neutrino masses • (inﬂation may be due to a separate sector) but give up the requirement of naturalness (at least do not require systematically naturalness). Might be “just so” or environmental selection (anthropic principle), or maybe we are not playing with the right parameters? Arkani-Hamed, Dimopoulos ’04 Split supersymmetry Giudice, Romanino ’04

Idea: abandon naturalness, which is in tension with flavour physics and proton decay, but keep the DM candidate and gauge coupling unification → split superpartner spectrum, with gauginos and higgsinos at the TeV scale and sfermions (as well as one of the two Higgs doublets) at a scale m˜ TeV 2 2 → proper EWSB is achieved by fine-tuning of order MZ /m˜ → gauge coupling unification by adjusting the gaugino mass

→ the DM particle is the lightestHigh-Scale SUSY neutralino (abundance SplitdeterminesSUSY µ)

10 10 ’14] Strumia Slavich, Giudice, [Bagnaschi, é é é é Lighter band: m 3 < mi < 3m Lighter band: m 3 < mi < 3m Darker: Mt = 173.34 ± 0.76 GeV darker: Mt= 173.34 ± 0.76 GeV

ê ê The Higgs mass constraint and the tuning

of the weak scale b determine (within b

tan 3 tan 3 uncertainties) m˜ and tan β Tuning condition for universal scalars

1 1 104 106 108 1010 1012 104 106 108 1010 1012 SUSY scale in GeV SUSY scale in GeV

Figure 4: Left: Regions in the (˜m, tan ) plane that reproduce the observed Higgs mass for High-

Scale SUSY. The black solid line gives the prediction for Xt =0, mass-degenerate superparticles, and central values for the SM parameters. The light-blue band shows the e↵ect of superparticle

thresholds by varying the supersymmetric parameters M1,M2,M3,mQi ,mUi ,mDi ,mEi , mLi and µ randomly by up to a factor 3 above or below the scale m˜ , and At within the range allowed by vacuum stability. The dark-blue band corresponds to mass-degenerate superparticles, but

includes a 1 variation in Mt. Right: Same as the left plot for the case of Split SUSY. The gaugino and higgsino masses are all set to 1TeV, and At =0. The dot-dashed curve corresponds to the EW tuning condition in the case of universal scalar masses at the GUT scale.

corrections that could reduce the Higgs mass when the parameter X˜ =(A µ cot )2/m m t t Q3 U3 is larger than about 12. The well-known bounds valid in the case of natural SUSY (see, e.g., ref. [35]) need to be adapted to the case of High-Scale SUSY, where the mass term for a combination of the two MSSM Higgs doublets almost vanishes because of the electroweak ﬁne-

tuning. In order to determine the upper bound on X˜t, let us consider the scalar potential for the stop-Higgs system

2 2 2 2 gt V = m Q˜ + m U˜ + A H Q˜ U˜ + µH⇤Q˜ U˜ +h.c. Q3 | 3| U3 | 3| sin t u 3 3 d 3 3 g2 ⇣ ⌘ + t H Q˜ 2 + H U˜ 2 + Q˜ U˜ 2 + Higgs-mass terms + D-terms , (37) sin2 | u 3| | u 3| | 3 3| ⇣ ⌘ where the appropriate SU(2)L contractions are implicit and where gt is the top Yukawa coupling of the SM. Let us consider the potential along the direction of the approximately-massless Higgs

ﬁeld H (with Hu = H sin , Hd = ✏H⇤ cos )andalongasquarkdirectionsuchthattheD-terms

19 [Bagnaschi, Giudice, Slavich, Strumia ’14] High-scale SUSY tuning condition Split SUSY tuning condition High-Scale SUSY CMSSMSplit SUSY, A = 0 0 m2 =m2 , negligible m, M 4 é Q U 1,2,3 10 10 m = 103 GeV 3.0 é é b = é é Lighter band: m 3 < mi < 3m tan 40. Lighter band: m 3 < mi < 3m é Darker: Mt = 173.34 ± 0.76 GeV H darkerm = :10M4t=GeV173.34L± 0.76 GeV H Lé m = 104 GeV tanb = 40. 2.5 é é 5 tanb = 21. ê 3 é ê m = 106 GeV m = 10 GeV m = 105 GeV é tanb = 1.8 tanb = 2.6 tanb = 3.8m = 106 GeV 2.0 tanb = 2.5 u H 2 2 m ê m b b d ê H 2

2 0 2 é 7 1.5 tan tan 3 3 m = 10 GeV m m

tanb = 2.0 = é 7

H m = 10 GeV Tuning condition r tanb = 1.4 for universal scalarsé 1.0 m = 108 GeV 1 tanb = 1.6 é m = 108 GeV é 0.5 tanb = 1.0 m = 109 GeV tanb = 1.4 1 01 0.0 4 6 8 10 12 4 6 8 10 12 10 10 10 10 10 0 10 1 10 2 10 103 104 0.0 0.5 1.0 1.5 2.0 SUSY scale in GeV SUSY scale in GeV 2 2 r = m2 m2 m1 2 m Q Q Hu

ê ê Figure 4: Left: Regions in the (˜m, tan ) plane thatFor reproduce universal the observed scalarê masses Higgs mass at forthe High- GUT scale, Figure 6: Left: Prediction of the SUSY-breaking scale m˜ and the value of tan from the EW Scale SUSY. The black solid line gives the prediction for X =06, mass-degenerate superparticles, tuning m ˜ condition t 10 GeV and the and Higgs tan mass, in2 High-Scale SUSY with universal gaugino mass m and central values for the SM parameters. The light-blue⇡ band shows the e↵ect of superparticle⇡ 1/2 and scalar mass m0 at the GUT scale (with A0 =0). The prediction is plotted as a function thresholds by varying the supersymmetric parameters M1,M2,M2 3,m2Qi ,mUi ,m2 Di2,mEi , mLi and of the ratios m1/2/µ and m0/µ evaluated at the GUT scale. The lines are truncated when µ randomly by up to a factor 3 above or below the scale m˜ , and At within the range allowed the vacuum-stability condition is violated. Right: same as in the left plot, in Split SUSY with by vacuum stability. The dark-blue band corresponds to mass-degenerate superparticles, but includes a 1 variation in M . Right: SameSU(5) as the relations left plot for for the the case scalar of Split masses. SUSY. The The prediction is plotted as a function of the ratios t 2 2 2 2 mQ/mH and mH /mH evaluated at the GUT scale. In the shaded region, the EW vacuum is gaugino and higgsino masses are all set to 1TeV, and Aut =0. Thed dot-dashedu curve corresponds to the EW tuning condition in the case of universalunstable. scalar masses at the GUT scale. the two-loop RGE of the MSSM. In figure 5 we show the minimum amount (in percent) corrections that could reduce the Higgs mass when the parameter X˜ =(A µ cot )2/m m by which one couplingt g ˆ (tM ) shouldQ3 beU changed3 in order to achieve an exact crossing is larger than about 12. The well-known bounds valid in the case of naturali GUT SUSY (see, e.g., gˆ (M )=ˆg (M )=ˆg (M )atsomeM , neglecting GUT-scale thresholds. The gray ref. [35]) need to be adapted to the case of High-Scale1 GUT SUSY,2 GUT where3 theGUT mass term for aGUT band is obtained by scanning the SUSY mass parameters by up to a factor 3 above or below combination of the two MSSM Higgs doublets almost vanishes because of the electroweak fine- the scalem ˜ ,andAt within the range allowed by vacuum stability, with tan adjusted so as to tuning. In order to determine the upper bound on X˜t, let us consider the scalar potential for the stop-Higgs system reproduce the measured value of the Higgs mass. For comparison, in the SM g2(MGUT)islarger than the value corresponding to perfect unification by approximately 3.5%. The figure shows 2 ˜ 2 2 ˜ 2 gt that in˜ High-Scale˜ ˜ SUSY˜ perfect gauge-coupling unification can still be achieved as long as the V = mQ3 Q3 + mU3 U3 + AtHuQ3U3 + µHd⇤Q3U3 +h.c. | | | | sin SUSY scalem ˜ is lower than a few times 106 GeV. 2 ⇣ ⌘ gt ˜ 2 ˜ 2 ˜ ˜ 2 + 2 HuQ3 + HuU3 + Q3U3 + Higgs-mass terms + D-terms , (37) sin | | | | | Tuning| of the EW scale ⇣ ⌘ where the appropriate SU(2)L contractions areThe implicit measurement and where ofgt is the the Higgs top Yukawa mass has coupling been a crucial new element for all schemes of High- of the SM. Let us consider the potential along theScale direction SUSY becauseof the approximately-massless it provides direct information Higgs (although blurred by the unknown parameter field H (with Hu = H sin , Hd = ✏H⇤ cos )andalongasquarkdirectionsuchthatthetan ) on the SUSY-breaking scalem ˜ .D-terms Moreover, although such unnatural schemes do not provide any dynamical explanation for the tuning of the EW scale, the very existence of the

19 21 Signature of split supersymmetry: long-lived gluino The gluino decays via virtual squarks:

2 TeV 5 m˜ 4 c⌧ = 0.4m g˜ M 107 GeV ✓ g˜ ◆ ✓ ◆

tan 1 decays outside detector (c⌧g˜ > 10 m) ' 1 . tan . 2 displaced vertex (c⌧g˜ > 50 µm) tan & 2 prompt decay High-scale supersymmetry

Now all superpartners have masses around m˜ TeV 2 2 → proper EWSB is achieved by ﬁne-tuning of order MZ /m˜ 6 → gauge coupling uniﬁcation can be achieved for m˜ . 10 GeV → no automatic DM candidate High-Scale SUSY Split SUSY 10 10 é é é é Lighter band: m 3 < mi < 3m ’14] Strumia Slavich, Giudice, [Bagnaschi, Lighter band: m 3 < mi < 3m Darker: Mt = 173.34 ± 0.76 GeV darker: Mt= 173.34 ± 0.76 GeV

The Higgs mass constraint provides ê ê a relation (within uncertainties) between m˜ and tan β b b

10 tan 3 tan 3 m˜ . 2 10 GeV for degenerate ⇥ Tuning condition superpartners and central SM parameters for universal scalars

For large tan β and large stop mixing, recover the several TeV region 1 1 104 106 108 1010 1012 104 106 108 1010 1012 SUSY scale in GeV SUSY scale in GeV

Figure 4: Left: Regions in the (˜m, tan ) plane that reproduce the observed Higgs mass for High-

Scale SUSY. The black solid line gives the prediction for Xt =0, mass-degenerate superparticles, and central values for the SM parameters. The light-blue band shows the e↵ect of superparticle

tuning. In order to determine the upper bound on X˜t, let us consider the scalar potential for the stop-Higgs system

ﬁeld H (with Hu = H sin , Hd = ✏H⇤ cos )andalongasquarkdirectionsuchthattheD-terms

19 [Bagnaschi, Giudice, Slavich, Strumia ’14]

High-Scale SUSY High-scale SplitSUSYSUSYtuning condition Split SUSY tuning condition CMSSM, A0 = 0 2 2 10 10 mQ=mU, negligible m, M1,2,3 4 é 3 é é m = 10 GeV é é 3.0 Lighter band: m 3 < mi < 3m tanb = 40. Lighter band: m 3 < mi < 3m Darker: Mt = 173.34 ± 0.76 GeV darkeré : Mt= 173.34 ± 0.76 GeV H m = 104 GeV L H Lé m = 104 GeV tanb = 40. 2.5 ê ê é 6 é 5 tanb = 21. 3 é m = 10 GeV m = 10 GeV m = 105 GeV é tanb = 1.8 tanb = 2.6 tanb = 3.8m = 106 GeV 2.0 tanb = 2.5 u H 2 2 m b b ê m d ê H 2 2 0 2 é 7 tan 3 tan 3 1.5

m = 10 GeV m m

tanb = 2.0 = é 7

H m = 10 GeV Tuning condition r tanb = 1.4 for universal scalarsé 1.0 m = 108 GeV 1 tanb = 1.6 é m = 108 GeV é 0.5 tanb = 1.0 m = 109 GeV tanb = 1.4 1 01 0.0 4 6 8 10 12 4 6 8 10 12 10 10 10 10 10 0 10 1 10 2 10 310 104 0.0 0.5 1.0 1.5 2.0 SUSY scale in GeV SUSY scale in GeV 2 2 r = m2 m2 m1 2 m Q Q Hu

For given GUT-scale ratios M /µ and m /µ , the Higgs massê constraint and ê Figure 4: Left: Regions in the (˜m, tan1/2) plane that reproduce0 the observedê Higgs mass for High- Figure 6: Left: Prediction of the SUSY-breaking scale m˜ and the value of tan from the EW the tuning of the weak scale determine m˜ and tan Scale SUSY. The black solid line gives the prediction for Xt =0,β mass-degenerate superparticles, tuning condition and the Higgs mass, in High-Scale SUSY with universal gaugino mass m1/2 and central values for the SM parameters. The light-blue band shows the e↵ect of superparticle and scalar mass m0 at the GUT scale (with A0 =0). The prediction is plotted as a function thresholds by varying the supersymmetric parameters M1,M2,M2 3,m2Qi ,mUi ,m2 D2i ,mEi , mLi and of the ratios m1/2/µ and m0/µ evaluated at the GUT scale. The lines are truncated when µ randomly by up to a factor 3 above or belowthe the vacuum-stability scale m˜ , and A conditiont within the is violated. range allowedRight: same as in the left plot, in Split SUSY with by vacuum stability. The dark-blue band correspondsSU(5) relations to mass-degenerate for the scalar superparticles, masses. The but prediction is plotted as a function of the ratios includes a 1 variation in Mt. Right: Same as2 the2 left plot for2 the2 case of Split SUSY. The mQ/mHu and mHd /mHu evaluated at the GUT scale. In the shaded region, the EW vacuum is gaugino and higgsino masses are all set to 1TeVunstable., and At =0. The dot-dashed curve corresponds to the EW tuning condition in the case of universal scalar masses at the GUT scale. the two-loop RGE of the MSSM. In figure 5 we show the minimum amount (in percent) 2 corrections that could reduce the Higgs mass when the parameter X˜t =(At µ cot ) /mQ mU by which one couplingg î(MGUT) should be3 changed3 in order to achieve an exact crossing is larger than about 12. The well-known bounds valid in the case of natural SUSY (see, e.g., gˆ1(MGUT)=ˆg2(MGUT)=ˆg3(MGUT)atsomeMGUT, neglecting GUT-scale thresholds. The gray ref. [35]) need to be adapted to the case ofband High-Scale is obtained SUSY, by where scanning the the mass SUSY term mass for a parameters by up to a factor 3 above or below combination of the two MSSM Higgs doubletsthe almost scale vanishesm ˜ ,and becauseAt within of the the range electroweak allowed fine- by vacuum stability, with tan adjusted so as to tuning. In order to determine the upper bound on X˜ , let us consider the scalar potential for reproducet the measured value of the Higgs mass. For comparison, in the SM g2(MGUT)islarger the stop-Higgs system than the value corresponding to perfect unification by approximately 3.5%. The figure shows that in High-Scale SUSY perfect gauge-coupling unification can still be achieved as long as the 2 2 2 2 gt V = m Q˜ + m U˜ + A H Q˜ U˜ + µH⇤Q˜ U˜ +h.c. 6 Q3 | 3| U3 | 3| sin SUSYt u scale3 3 m ˜ isd lower3 3 than a few times 10 GeV. g2 ⇣ ⌘ + t H Q˜ 2 + H U˜ 2 + Q˜ U˜ 2 + Higgs-mass terms + D-terms , (37) sin2 | u 3| | u 3| | Tuning3 3| of the EW scale ⇣ ⌘ The measurement of the Higgs mass has been a crucial new element for all schemes of High- where the appropriate SU(2)L contractions are implicit and where gt is the top Yukawa coupling of the SM. Let us consider the potential along theScale direction SUSY becauseof the approximately-massless it provides direct information Higgs (although blurred by the unknown parameter tan ) on the SUSY-breaking scalem ˜ . Moreover, although such unnatural schemes do not field H (with Hu = H sin , Hd = ✏H⇤ cos )andalongasquarkdirectionsuchthattheD-terms provide any dynamical explanation for the tuning of the EW scale, the very existence of the

21 19 Scenarios with superheavy sterile neutrinos (seesaw)

An economical way to generate both neutrino masses and the BAU: add RH neutrinos to the SM with a large Majorana mass term 1 I = Y N L H M N N c +h.c. Lseesaw i↵ Ri ↵ 2 i Ri Ri → light (Majorana) neutrino masses from heavy Majorana neutrino exchange

YiYj⇥ 2 ⇒ (M⇤ )⇥ = v M i i

14 2 for M 10 GeV , get m ⌫ = m from Y 1 ⇠ atm ⇠ q CP violation: being→ MajoranaBAU from fermions, the lepton the asymmetry heavy neutrinos generated are in their CP-violating, own out-of- antiparticles and equilibriumcan decay bothdecays into of thel⁺ and heavy into Majorana l⁻ neutrinos (leptogenesis)

! Γ(Ni → LH) =" Γ(Ni → LH¯ ) at the 1-loop level

The decay rates into l⁺ and into l⁻ differ due to quantum corrections

! ⇒ Γ(Ni → LH) =" Γ(Ni → LH¯ ) ⇒ asymmetry between lepton and antilepton abundances, which is partially washed out by L-violating processes and converted into a baryon asymmetry by the sphalerons In principle, SM neutrino masses could be generated from TeV-scale RH neutrinos (at the price of unnaturally small Yukawa couplings), but successful leptogenesis requires heavy Majorana neutrinos

SM MSSM 1016 1016 M ≥ . − . × 9 hep-ph/0310123] [Giudiceal., et 1 (0 5 2 5) 10 GeV 1014 1014 depending on the initial conditions

12 12 [Davidson, Ibarra] 10 zero N1 10 zero N1 GeV GeV in in

1 1

N 10 N 10

Way out: “resonant leptogenesis” m 10 m 10

thermal N1 thermal N1 M 1 TeV possible at the price of 108 108 ⇠ dominant N1 dominant N1 a strong tuning M1 M2 ' 106 106 [Covi, Roulet, Vissani - Pilaftsis] 10−10 10−8 10−6 10−4 10−2 1 10−6 10−4 10−2 1 ∼ ∼ m1 in eV m1 in eV

Barring this possibility, no experimental signature at low energy Figure 9: Allowed range of m˜ 1 and mN1 for leptogenesis in the SM and MSSM assuming No DM candidate either. Most economicalm3 = mpossibilityax(m˜ 1, matm )isa nand ξ axion,= m3/m˜ which1. Succe salsosful le ptogenesis is possible in the area inside solves the strong CP problem the curves (more likely around the border).

Interestingly, adding an axion can renderIn ftheact, eEWven if vacuumN1 initially absolutelyhas a therma lstableabundan upcy ρ /ρ g /g 1, its contribution N1 R ∼ N1 ∗ " to the Planck scale, allowing for Higgs inflationto the total den[Salviosity of ’15]the universe becomes no longer negligible, ρN1 /ρR (gN1 mN1 )/(g!T ), if it decays strongly out of equilibrium at T m . For the reasons ex∼plained above, this " N1 effect gives a suppression of η (rather than an enhancement), and for very small m˜ 1 the case (1) and ( ) give the same result. ∞ The lower panel of fig. 8 contains our result for the efficiency η of thermal leptogenesis | | 14 computed in cases (0), (1) and ( ) as function of both m˜ 1 and mN1 . At mN1 > 10 GeV non-resonant ∆L = 2 scatterin∞gs enter in thermal equilibrium strongly sup∼pressing η. Details depend on unknown flavour factors. Our results in fig. 8 can be summarized with simple analytical fits

−3 1.16 1 3.3 10 eV m˜ 1 × + −3 in case (0) (40) η ≈ m˜ 1 0.55 10 eV ! × " 14 valid for mN1 10 GeV. This enables the reader to study leptogenesis in neutrino mass models withou"t setting up and solving the complicated Boltzmann equations.

Implications

Experiments have not yet determined the mass m3 of the heaviest mainly left-handed neutrino. We assume m3 = max(m˜ 1, matm). Slightly different plausible assumptions are possible when m˜ 1 matm, and very different fine-tuned assumptions are always possible. ≈ 20 Can one test leptogenesis? In full generality, no. Requires CP violation in the lepton sector, but in general depends both on low-energy and high-energy phases Nevertheless, would gain support from: - observation of neutrinoless double beta decay [proof of the Majorana nature of neutrinos - necessary condition] - observation of CP violation in the lepton sector, e.g. in neutrino oscillations - experimental exclusion of non-standard electroweak baryogenesis scenarios [e.g. MSSM with a light stop or other extensions of the SM] Theoretical expectations/predictions

Many new physics scenarios predict “large” LFV rates: supersymmetry, extra dimensions, little Higgs models, ... In (R-parity conserving) supersymmetric extensions of the Standard Model, LFV is induced by a misalignment between the lepton and slepton mass matrices, parametrized by the mass insertion parameters (α ≠ β): (m2 ) 2 e LL L˜ ⇥ RR (me˜)⇥ RL A⇥vd ⇥ 2 , ⇥ 2 , ⇥ mL mR mRmL AIn more the mass optimistic insertion conclusion approximation, holds the for branching the supersymmetric ratio for µ → seesaw e γ reads 3 mechanism: the RH3⌃ neutrino couplings induce2 ﬂavour violation in2 the BR (µ e⇤)= f ⌅LL + f ⌅LR + f ⌅RR + f ⌅LR tan2 ⇥ slepton sector through2 4 radiativeLL 12 correctionsLR 12 RR 12 LR 21 4GF cos ⇧W ⇥ ⇤ →with lepton fL, fR functionsﬂavour violating of the superpartnerprocesses such masses as µ → and e γof, τtan → βµ. Forγ, µ moderate→ eee, to τlarge → µµµ tan , βµ, -thee conversion branching ratioin nuclei approximately scales as tan² β

4 5 MW LL 2 2 ⇒ typical µ → e γ rate: B(µ e⇥) 10 4 ⇤12 tan ⇥ MSUSY | | 13 to be compared with the experimental limit 5 . 7 10 (MEG 2013), which ⇥ should improve further by 1 order of magnitude Other expected improvements: µ → eee (Mu3e proposal, 4 orders of magn.), µ-e conversion in nuclei (DeeMee, 2 orders of magn.; Mu2e and COMET, 4 orders of magn.) details of our numerical analysis. In section 4, we present our results. We conclude with a summary and outlook in section 5. Finally, in appendix A we describe the proposed future experiments and their expected sensitivity.

2 Seesaw in mSUGRA and NUHM

The phenomenology of SUSY Type I seesaw mechanism with universal boundary conditions (mSUGRA/CMSSM) has been studied in many papers (see [29, 30] for a set of recent works). Here we review some essential features related to flavor violation for completeness and to do a comparison with the case of non-universal Higgs masses. To set the notation, the Type I seesaw mechanism is characterized by a superpotential containing the following terms 1 Y LecH + Y L⌫cH + M ⌫c⌫c (2.1) W e d ⌫ u 2 R where L (ec) stands for the leptonic doublets (singlets) and ⌫c are the right-handed (RH) neutrino superfields (with the generation indices not explictely written). Ye and Y⌫ are the electron and neutrino (Dirac) Yukawa matrices. In models like CMSSM/mSUGRA, the soft terms are assumed to be universal at the Grand Unification (GUT) scale, M 2 1016 GeV. At the weak scale as is well known, GUT ⇠ ⇥ the soft terms are no longer universal due to the e↵ects of the renormalization group (RG) running. The presence of the RH neutrinos of eq. (2.1) at an intermediate scale contribute to the running and generate flavor violating entries in the left-handed slepton mass matrix at the weak scale [31]. At the leading order these terms can be estimated to be:

2 2 2 ` 3m0 + A0 MX (mL˜)i=j i=j 2 (Y⌫⇤)ik (Y⌫)jk log , (2.2) 6 ⌘ 6 LL ⇡ 8⇡ MRk ⇣ ⌘ Xk ✓ ◆ th where MX represents the GUT scale and MRk , the scale of the k RH neutrino. m0 and A0 stand for the usual universal soft mass and trilinear terms at the high scale. Y⌫, the Dirac neutrino Yukawa couplings are free parameters in the Type I seesaw mechanism which cannot be completely determined even after including the complete data on the neutrino mass matrix [32]. SO(10) models with their matter representations being 16-dimensional provide a natural setting for the seesaw mechanisms. Furthermore, they provide information about the neutrino Yukawa couplings. For example, it is known that as long as we restrict to renor- malisable SO(10) models, at least one of the neutrino Yukawa couplings should be as large as the top Yukawa coupling [25]. Thus with suitable assumptions for the (left-handed) explicit supersymmetricmixing of the Dirac seesaw Yukawa models Neutrino lead mass to matrix, testable one can predictions make predictions for the ﬂavor violation generated at the weak scale from eq. (2.2). Two extreme scenarios for mixing are Example: SO(10)-inspiredtypically considered mass to relations be present Y in =Y Yu⌫ [ 25 [Calibbi, 26, 33 ]:et al. ’12] assume equal eigenvalues Y⌫ = Yu (CKM Case) (blue) diag but allow different mixings Y⌫ = Yu UPMNS (PMNS Case), (red) (2.3)

diag where Yu = VCKMYu VCKM† . Both these scenarios can be motivated from concrete models of fermion masses within the SO(10) framework [25, 26]. The ﬂavor violating o↵- diagonal entries at the weak scale, eq. (2.2), are then completely determined by assuming

–3–

Figure 3. Thescan figure over in the mSUGRA left panel showsparameters, the BR(µ tan eβ )= obtained 10, Ue3 by = scanning0.11 the mSUGRA ! parameters in the ranges given in eq. (3.1) and for fixed tan = 10 and Ue3 =0.11 (the lowest value allowed at 3 by recent RENO observation). The red (blue) colored points correspond to PMNS (CKM) case. Di↵erent horizontal lines correspond to present and future bounds on BR(µ e). ! The figure in the right panel shows the allowed space in the m0 m1/2 plane which satisfy the current MEG bound. The region below the red line is excluded by the current LHC searches [7]. Both the plots satisfy all the constraints in eq. (3.2).

suppressed branching fractions due to the smallness of CKM angles (see table (1)) as has been detailed in [26]. Though there has been no strong improvements in the experimental sensitivity compared to the analyses of [26], we update the result with the light Higgs mass constraint. In figure 3 we show the results for tan = 10. As we can see, some part of the parameter space of the CKM case can be probed by the proposed Project-X experiment4 for µ e. At present the main constraint to this scenario is simply provided by the m ! h range of eq. (1.1), that excludes the regions with lighter SUSY spectra: m0 . 2 TeV for small M1/2, M1/2 . 1 TeV for small m0, as we can see from the right panel of the figure. We can also notice that the LHC limits on the mSUGRA parameter space has already started to constrain regions of the parameter space otherwise allowed by the bounds in eq. (3.2). Let us now turn our attention to other observables like µ eee, µ e conversion ! ! in nuclei and ⌧ µ, which is independent of ✓ . In figures 4, 5 and 6,weshowthe ! 13 predicted rates for ⌧ µ, µ eee and µ e conversion in the Titanium nucleus versus ! ! ! the BR(µ e) (that is at present the most constraining LFV observable), for the PMNS ! case in mSUGRA (red points) and in NUHM1 (green points) as well as for the CKM case (blue points). As can be seen from figure 4, in the PMNS case, the present MEG limit on BR(µ e) ! implies BR(⌧ µ) 10 12, beyond the reach of the proposed experiments. This is a ! . direct consequence of the large value of ✓13 measured by Daya Bay and RENO. In fact, from eq.(2.4,2.5) and table 1,wehave:

BR(⌧ µ) U U 2 ! | ⌧3 µ3| BR(⌧ µ⌫⌫¯) (1). (4.1) BR(µ e) ⇡ U U 2 ⇥ ! ⇡ O ! | µ3 e3|

In the CKM case (blue points), the small mixing angle and the mh bound are such that BR(⌧ µ) 10 10. Thus, the scenarios discussed here allow possible signals of LFV in ! . 4In appendix A we present a brief summary of all the future experimental facilities related to the ﬂavor violating observables discussed in the text.

–8– A more predictive scenario is provided by the supersymmetric type II seesaw model, in which a scalar electroweak triplet generates small neutrino masses and the BAU through leptogenesis Correlations between µ → e γ and other LFV processes

— τ → µ γ — µ Ti → e Ti — µ → e e e

Calibbi, Frigerio, SL, Romanino ’09

(triplet parametersCorrelations ﬁxed,between scan µ → overe γ and m other 0 < LFV3 TeV processes ,M 1 / 2 < 2 TeV , ) (triplet parameters ﬁxed, scan over m A 0 0< =03 TeV , ,M tan 1 / 2 < = 2 TeV10 ,µ> , 0

A 0 =0 , tan = 10 ,µ> 0 ) Scenario with keV/GeV sterile neutrinos: the νMSM Asaka, Blanchet and Shaposhnikov ’05; Asaka and Shaposhnikov ’05

Drastic approach: in addition to the hierarchy problem, abandon naturalness of the SM neutrino masses Just add 3 RH neutrinos with Majorana mass terms and small Yukawa couplingsSterile to the SM neutrinos neutrinos as dark matter → SM neutrino masses from the seesaw mechanism → dark matter4 Sterile provided neutrinos are by fermions N1 with and mass obey inthe the exclusion few keVprinciple. range It is not possible to have an arbitrarily large ns number density. → BAU from The N observed2 and NDM3 withdensity masses in dwarf in galaxies the (1-100) implies aGeV lower range limit on the DM mass. The sterile neutrino DM N1 is produced by active-sterile oscillations in the early Universe.5 Sterile It neutrinoscan decay are notas N absolutely ⌫ stable ⇒ X-ray constraints 1 ! i

Sterile neutrino and 3.5 keV line

10-6 Too much Dark Matter

)] -7

θ 10 Non-resonant production Excluded by non-observation (2

2 of dark matter decay line 10-8 Lyman-α bound -9 L for NRP sterile neutrino 10 6 =12

10-10 L 6 =25 Boyarski -11 L 10 6=70 BBN limit: L L max 10-12 6 =700

constraints BBN 6 = 2500 10-13 Phase-space density Interaction strength [Sin 10-14 Not enough Dark Matter 1 5 10 50 DM mass [keV]

Very small mixing angle: N1 practically decoupled from the SM neutrinos Can be detected from its decays (X-ray line) Alexey Boyarsky DEEPER VIEW ON NEUTRINO AS DARK MATTER 37 0.1 BBN Seesaw N Seesaw I 0.001 BAU N " s

! BAU I BAU not generated in out-of-equilibriumΤ 10!5 decays of N2 and N3, but from a different mechanism involving active-sterile neutrino oscillations 10!7 before the electroweak phase transition [Akhmedov, Rubakov, Smirnov ’98] 10!9 0.2 0.5 1.0 2.0 5.0 10.0 M !GeV" Requires a tuning M2 M3 Figure' 2: Constraints on the HNL lifetime, ⌧ , from Big Bang Nucleosynthesis (black line: “BBN”), from the baryon asymmetry of the Universe (“BAU”) and from the seesaw mechanism (blue solid lines: “BAU N” and “Seesaw N” refer to a normal mass-hierarchy of active neutrinos and “BAU I” and “Seesaw I” refer to an N2 and N3 can be producedinverted mass-hierarchy). in Themeson allowed region decays of the parameter through space is shown in white their for the normal mixing hierarchy with active neutrinos case. The limits from direct experimental searches are outlined in Fig. 4. Figure taken from Ref. [45]. µ Ds µ νµ N 2,3 νµ H N2,3 H υ υ π µ D µ N2,3 νµ ν π µ H e H N ν υ 2,3 υ e

Figure 3: Feynman diagrams (left) for the production of HNLs and (right) for their decays. The dashed line denotes the coupling to the Higgs vacuum expectation value, leading to the mixing of active neutrinos and ⇒ dedicated experimentHNLs via (SHIP) Yukawa couplings. proposed at the CERN SPS [arXiv:1310.1762]

+ always below 2%. The µ⇡ ﬁnal state is the cleanest signature experimentally and is the focus of + + the studies below. The µ⇢ and e⇡ ﬁnal states provide additional experimental signatures that extend the sensitivity and could be used to constrain additional parameter space. 12 4 Assuming a branching fraction 10 and a factor 10 from the lifetime, an experiment to detect 16 N2,3 would require more than 10 D mesons in order to fully explore the parameter space with M<2 GeV. Preliminary studies of an experimental design were described in Ref. [46].

3 Experimental status and cosmological constraints

The region of the lifetime-mass (⌧ ,MN ) plane consistent with the cosmological constraints is shown 2 in Fig. 2. Figure 4 shows the allowed region in the (U ,MN ) plane, given the constraints from particle physics experiments. For all points in Fig. 4 below the line marked “Seesaw”, the mixing of the HNL with active neutrinos becomes too weak to produce the observed pattern of neutrino ﬂavour oscillations. Cosmological considerations result in additional limits. If the HNLs are required

5 N2 and N3 can also be produced in Z decays and then decay to ll’ν,

Normal hierarchylqq’ or qq¯ν 10-4 CHARM 10-5 Delphi 10-6 NuTeV -7 10 PS191 2 10-8 BAU |U|

10-9 Normal hierarchyFigure 4: Decay modes of heavy neutrinosInverted through hierarchy 10-6 10-6 mixing with light neutrinos: the charged currentNuTeV decay 10-10 NuTeV BBN N `⌫ (a), the neutral current decayCHARMN ⌫ + /Z. 10-7 ! 10-7 ! 10-11 PS191 PS191 Seesaw BAU -12 BAU 10 10-8 10-8 1 10 The production and decay of the heavy neutrino in

HNL mass2 (GeV) SHiP 2 -9 Z decays has already beenRegion studied-9 of at LEPsensitivitySHiP by the L3 at FCC-ee

|U| 10 |U| 10 Inverted hierarchy and DELPHI collaborations [32, 33]. It is largely de- 10-4 10-10 BBN FCC-eetermined by the mixing angle.10-10 WhenBBN a left-handed neu-FCC-ee 10-5 Delphi trino is produced e.g. in Z decays it is actually a mixture CHARM 10-11 10-11 Seesaw 10-6 Seesaw of the light and heavy state:

10-7 1 10 ⌫L = ⌫ cos ✓ + N sin ✓ 1 10 BAU HNL mass (GeV) HNL mass (GeV)

2 NuTeV 10-8

|U| 2 [Blondel, Graverini, Serra, Shaposhnikov, arXiv:1411.5230] PS191 (a) Decay length 10-100 cm,with 1012✓Z0 m⌫/mN . Thus the decay width(a) Decay of length the Z 10-100into a cm, 1012 Z0 -9 ⇡ 10 Normal hierarchypair of light and heavy neutrinos will be givenInverted by hierarchy 10-6 10-6 10-10 NuTeV NuTeV 2 BBN = SM 2 / CHARM2 + / 2 10-11 10-7 Z ⌫N 3 Z ⌫⌫ U 1 (m10N-7 mZ) 1 (mN mZ) Seesaw PS191 ! ! | | PS191 -12 BAU 10 BAU2 2 ⇣ ⌘ ⇣ ⌘ 10-8 with U ✓ . The best10 existing-8 limits are around 1 10 | | ⇠ HNL mass (GeV) 2 < 5 2 SHiP U 10 in the mass range2 relevant to high energy -9 | | -9 SHiP |U| 10 investigations (Figure 3). The|U| 10 mixing of sterile neutri- Figure 3: Interesting domains in the mass-coupling pa- nos with the active neutrinos of each flavour i is defined 10-10 BBN FCC-ee 2 10-10 BBN 2 FCC-ee rameter space of heavy neutrinos and current experi- as Ui , where i = e, µ, ⌧. The total mixing U is de- mental limits, for normal and inverted hierarchy of the | | 2 2 | | -11 fined as U = Ui . The measurement-11 of the partial 10 | | i | | 10 Seesaw left-handed neutrino masses. Seesaw width is sensitive to U 2, while in direct searches the P | | 1 final10 state depends on the relative strength1 of the partial 10 HNL mass (GeV) 2 HNL mass (GeV) Ui . In our analysis we consider the combination of | |2 U 13allowed0 by present constrains from neutrino oscil- 13 0 neutrinos with active neutrinos. Below(b) Decaythis line, length the 10-100 cm,| i 10| Z (b) Decay length 10-100 cm, 10 Z active neutrino mass di↵erences observed in neutrinoNormal hierarchylations that maximises the BAU. Inverted hierarchy 10-6 Heavy neutrinos decay as10 shown-6 in Figure 4. At large experiments cannot be accounted for in the GeV scaleNuTeV NuTeV masses the fully visible decay NCHARM`W(W qq) ac- see-saw mechanism. Above the BAU-7 line the reactions -7 10 PS191 10 ! ! with right-handed neutrinos are in thermal equilibrium counts for more than 50% of thePS191 decays. BAU BAU during the relevant period of the10-8 Universe expansion, The decay rate of the heavy10-8 neutrino depends very making the baryogenesis due to right-handed neutrino strongly on the mass, both via the three body phase 2 SHiP 2 -9 -9 SHiP oscillations impossible. For mN|U| 10close to MW and above space (in the fifth power of|U| 10 mass) but also through the MW the rate of reactions with N’s is enhanced due to mixing angle. The average decay length is given in [32]: -10 BBN -10 BBN the kinematically allowed decay10 N `W leading to 10 ! FCC-ee 3[cm] FCC-ee stronger constrains on the mixing [30]. The BAU curve L 10-11 2 10-11 6 Seesaw Seesaw ⇠ U (mN [GeV]) intersects with the see-saw line at mN = MW , so that the | | ⇥ parameter space is bound on all sides. 1 10The existence of heavy neutrinos in the1 accessible 10 HNL mass (GeV) HNL mass (GeV) For even larger masses of N another mechanism of mass range would manifest itself in many di↵erent ways baryogenesis – resonant leptogenesis –(c) can Decay operate length [31]. 0.01-500 cm,in high 1013 Z energy0 colliders. (c) Decay length 0.01-500 cm, 1013 Z0 This part of the parameter space cannot be directly stud- If N is heavy but within kinematical reach, it will de- ied with the FCC-ee operatedFigure at the 8:Z Regionsresonance. of sensitivity forcay sterile in neutrinos the detector as a and couldFigure be 9: detectedRegions of and sensitivity possibly for sterile neutrinos as a function of mass and mixing to light neutrinos (normal function of mass and mixing to light neutrinos (inverted hierarchy): for 1012 Z decays occurring3 between 10 cm hierarchy): for 1012 Z decays occurring between 10 cm and 1 m from the interaction point (a), same for 1013 Z and 1 m from the interaction point (a), same for 1013 Z decays (b), for 1013 Z decays occurring between 100 µm decays (b), for 1013 Z decays occurring between 100 µm and 1 m from the interaction point (c). and 1 m from the interaction point (c). 6