Aspects of Symmetry Breaking in Grand Unified Theories

Home , Chirality (physics), Split supersymmetry

arXiv:1110.3210v1 [hep-ph] 14 Oct 2011 r tfn Bertolini Stefano Dr. Supervisor: set fSmer raigin Breaking Symmetry of Aspects coaItrainl ueir iSuiAvanzati Studi di Superiore Internazionale Scuola rn nﬁdTheories Uniﬁed Grand hsssbitdfrtedge of degree the for submitted Thesis ret,Spebr2011 September Trieste, otrPhilosophiae Doctor SISSA uaD Luzio Di Luca Candidate: 2 Abstract

We reconsider the issue of spontaneous symmetry breaking in (10) grand unified theories. The emphasis is put on the quest for the minimal Higgs sector leading to a phenomenologically viable breaking to the standard model gauge group. Longstanding results claimed that nonsupersymmetric SO(10) models with just the adjoint representation triggering the first stage of the breaking cannot provide a successful gauge unification. The main result of this thesis is the observation that this no-go is an artifact of the tree level potential and that quantumSO corrections opens in a natural way the vacuum patterns favoured by gauge coupling unification. An analogous no-go, preventing the breaking of (10) at the renormalizable level with representationsSO up to the adjoint, holds in the supersymmetric case as well. In this respect we show that a possibleSO way-out is provided by considering the flipped (10) embedding of the hypercharge. Finally, the case is made for the hunting of the minimal (10) theory. 4 Contents

Foreword 9

1 From the standard model to SO 19

(10) 1.1 Thestandardmodelchiralstructure ...... 20 1.2 TheGeorgi-Glashowroute...... 21 1.2.1 Charge quantization and anomaly cancellation ...... 23 1.2.2 Gaugecouplingunification ...... 24 1.2.3 Symmetrybreaking ...... 25 1.2.4 Doublet-Tripletsplitting ...... 28 1.2.5 Protondecay...... 29 1.2.6 Yukawasectorandneutrinomasses ...... 30 1.3 ThePati-Salamroute ...... 32 1.3.1 Left-Rightsymmetry ...... 33 SO1.3.2 Leptonnumberasafourthcolor ...... 38 1.3.3 Onefamilyunified ...... 40 1.4 (10)grouptheory ...... 40 1.4.1 Tensorrepresentations ...... 41 1.4.2 Spinorrepresentations...... 42 1.4.3 Anomalycancellation ...... 48 1.4.4 Thestandardmodelembedding.SO ...... 48 1.4.5 TheHiggssectorH ⊕ H ...... 49 1.5 Yukawa sectorH ⊕ inH renormalizable (10)...... 52 1.5.1 10 126 withsupersymmetry ...... 55 1.5.2 10 126 withoutsupersymmetry ...... 58 1.5.3d Type-Ivstype-IIseesaw ...... 60 1.6 Protondecayd ...... 61 1.6.1 d =6(gauge)...... 62 1.6.2 d =6(scalar)...... 64 1.6.3 =5...... 66 1.6.4 =4...... 68 Contents 26 Intermediate scales in nonsupersymmetric SO unification 71 SO (10) 2.1 Three-step (10)breakingchains ...... 72 2.1.1 Theextendedsurvivalhypothesis...... 73 2.2 Two-loop gauge renormalization groupU equations ...... 75 2.2.1 Thenon-abeliansector ...... 75 2.2.2 The abelian couplings and (1)mixing ...... 76 2.2.3 Somenotation...... 78 2.2.4U One-loopmatching...... R ⊗ U B−L 79 2.3 Numericalresults ...... 81 2.3.1 (1) φ (1) mixing ...... 82 2.3.2 Two-loopresults(purelygauge)126 ...... 82 2.3.3 The Higgsmultiplets...... 88 2.3.4 Yukawaterms ...... 91 3 The quantum vacuum of the minimal GUT 95 2.3.5 TheprivilegeofbeingminimalSO ...... 92

(10) 3.1 TheminimalSO(10)Higgssector ...... 95 3.1.1 Thetree-levelHiggspotential ...... 96 3.1.2 Thesymmetrybreakingpatterns ...... 97 3.2 Theclassicalvacuum ...... 100 3.2.1 Thestationarityconditions ...... 100 3.2.2 Thetree-levelspectrum ...... 101 3.2.3 Constraints ona the potential parameters ...... 101 3.3 Understandingthescalarspectrumλ ...... 102 2 3.3.1 45 only with =0...... 102a λ β τ 2 3.3.2 16 only with =0...... 102β τ 2 2 3.3.3 A trivial 45-16 potential ( = = = =0)...... 103 3.3.4 A trivialχR 45-16 interaction ( = =0) ...... 103 3.3.5 Atree-levelaccident ...... 103 3.3.6 The =0limit...... 105 3.4 Thequantumvacuum...... 105 3.4.1 Theone-loopeﬀectivepotential ...... 105 3.4.2 Theone-loopstationaryequations ...... 106 3.4.3 Theone-loopscalarmass...... 107 3.4.4 One-loopPGBmasses...... 108 4 SUSY- breaking with small representations 113 3.4.5SO Theone-loopvacuumstructure ...... 110

(10) SO 4.1 Whatdoneutrinostellus? ...... 113 4.2 SUSY alignment: a case for ﬂipped (10) ...... 116 4.3 TheGUT-scalelittlehierarchy ...... 118 Contents 7

4.3.1 GUT-scale thresholdsand protondecay...... 119 4.3.2 GUT-scaleSO thresholds and one-step unification ...... 120 4.3.3 GUT-scale thresholds and neutrino masses ...... 120 4.4 Minimal flipped (10)Higgsmodel ...... 120 4.4.1 Introducingthemodel...... 122 4.4.2 SupersymmetricvacuumE ...... 127 4.4.3Y Doublet-TripletB−L splittingE in flipped models ...... 129 6 4.5 Minimal Eembedding...... 131 6 4.5.1 and into ...... 134SU 6 4.5.2 The vacuummanifoldE ...... 134 4.5.3 Breaking the residual (5) via effective interactions ...... 138 6 4.5.4 A unified scenario ...... SO 139 4.6 Towardsarealisticflavor...... 140 Outlook:4.6.1 the quest Yukawa for sector the minimal of the flipped nonsupersymmetric(10)modelSO . .. ..theory ...... 147140

A One- and Two-loop beta coeﬃcients (10) 153 U

A.1 Beta-functions with (1)mixing ...... 159 B A.2SO Yukawacontributionsalgebra representations ...... 165 161

(10) B.1 TensorialrepresentationsC ...... 165 B.2 Spinorialrepresentations...... 166 B.3 The charge conjugation ...... 167 CB.4 Vacuum TheCartangenerators stability ...... 171 168 D Tree level mass spectra 173

D.1 Gaugebosons...... 173 D.1.1 Gaugebosonsmassesfrom45 ...... 174 D.1.2 Gaugebosonsmassesfrom16 ...... 174 D.2 Anatomyofthescalarspectrum ...... 175 D.2.1 45only...... χR 6 175 D.2.2 16only...... a λ β τ 175 D.2.3 Mixed 45-16 spectrum ( =0)β τ ...... 176 2 2 D.2.4 A trivial 45-16 potential ( = χR= = =0)...... 178 D.2.5 A trivial 45-16 interaction ( = =0) ...... 178 D.2.6 The 45-16 scalar spectrum for =0 ...... 179 Contents E8 One-loop mass spectra 183

E.1 GaugecontributionstothePGBmass ...... 183 F Flipped SO vacuum 189 E.2 ScalarcontributionstothePGBmass...... 184 SO (10) F.1 Flipped (10)notation...... 189 F.2 Supersymmetricvacuummanifold...... 191 F.3 Gaugebosonspectrum ...... 198 F.3.1 Spinorialcontribution ...... 199 F.3.2 Adjointcontribution ...... 200 G E vacuum 203 F.3.3 Vacuumlittlegroup...... 201 SU 6 E 3 G.1 The (3) formalism...... 203 6 G.2 vacuummanifold...... 206 G.3 Vacuumlittlegroup ...... 208 Foreword

This thesis deals with the physics of the 80’s. Almost all of the results obtained here could have been achieved by the end of that decade. This also means that the field of grand unification is becoming quite old. It dates back in 1974 with the seminal papers of Georgi-Glashow [1] and Pati-Salam [2]. Those were the years just after the foundation of the standard model (SM) of Glashow-Weinberg-Salam [3, 4, 5] when simple ideas (at least simple from our future perspective) seemed to receive an immediate confirmation from the experimental data. Grand unified theories (GUTs) assume thatSU allC the⊗U fundamentaQ l interactions of the SM (strong and electroweak) have a common origin. The current wisdom is that we live in a broken phase in which the world looks (3) (1) invariant to us and the low- energy phenomena are governed bySU strongL ⊗ interactions U Y and electrodynamics. Growing withU the energyQ we start to see the degreesU Q Ï of freedom SU L ⊗ of U a newY dynamics which can be interpreted1 as a renormalizable (2) SU(1) gaugeC ⊗ SU theoryL ⊗ spontaneously U Y broken into (1) . Thus,G in analogy to the (1) (2) (1) case, one can imagine that at higher energies the SM gauge group (3) (2) (1) is embedded in ′ Ma simpleU ≫ M groupW . G g g g SU TheC ⊗ first SU implicationL ⊗ U Y of the grand unification ansatz is thatgU at some mass scale ′ 3 2 g theg relevantg symmetry is and the , and coupling constants of (3) (2) (1) merge into a single gauge coupling . The rather different 3 2 values for , and at low-energy are then due to renormalization effects. Actually one of the most solid hints in favor of grand unification is the fact that the running within15 the SM shows an approximate convergence of the gauge couplings around 10 GeV (see e.g. Fig. 1). This simple idea, though a bit speculative, may have a deep impact on the understanding of our low-energy world. Consider for instance som2e unexplained features of the SM like e.g. charge quantization or anomaly cancellation . They appear just as the 1 At the time of writing this thesis one of the main ingredients of this theory, the Higgs boson, is still missing experimentally. On the other hand a lot of indirect tests suggest that the SM works amazingly well and it is exciting that the mechanism of electroweak symmetry breaking is being tested right now at the2 Large Hadron Collider (LHC). In the SM anomaly cancellation implies charge quantization, after taking into account the gauge invariance of the Yukawa couplings [6, 7, 8, 9]. This feature is lost as soon as one adds a right-handed Foreword -1 10 Αi 60 50 40 30 20 10

log10HΜGeVL 5 10 15 18

U Y G

One-loop running of the SM gauge couplings assuming the (1) embedding into . Figure 1: SO natural consequence of starting with an anomaly-free simple group such as (10). Most importantly grand unification is not just a mere interpretation of our low- energy world, but it predicts new phenomena which are correlated with the existing ÷ ones. The most prominentτp of& these is the instability of matter. The current lower bound on the proton lifetime is something33 34 like 23 orders of magnitude bigger than the age of the Universe, namely 10 yr depending on the decay channel [11]. This number is so huge that people started to consider baryon number as an exact symmetry of Nature [12, 13,3 14]. Nowadays we interpret it as an accidental global symmetry of the standard model . This also means that as soon as we extend the SM there is the chance to introduce baryon violating interactions. Gravity itself could be responsible for the breaking of baryon number [17]. However among all the possible frameworks there is only one of them which predicts a proton lifetime close to its experimental limitMU and this theory is grand unification. Indeed we can roughly estimate it by dimensional arguments. The exchange of a baryon-number-violating− MU vector boson of mass τp ∼αU ; yields something like mp4 1 − 5 αU ∼ τp & (1) MU & 1 33 and by putting in numbers15 (we take 40, cf. Fig. 1) one discovers that 10 yr corresponds to 10 GeV, which is consistent with the picture emerging in Fig. 1. ÷ NoticeMU that the gauge running is sensitive to the log of the scale. This means that a 10% variation on the gauge couplings at the electroweak scale induces a15 100%18 one on . Were the apparent unification of gauge couplings in the window 10 GeV just νR νR an accident, then Nature would have played a bad trick on us. neutrino3 , unless is a MajoranaSU particleL [10]. B L − π/α − In the SM the baryonic current is anomalous and baryone number∼ e violation can arise from instanton 2 transitions between degenerate (2) vacua which lead to ∆2 = ∆ =173 3 interactions for three flavor families [15, 16]. The rate is estimated to be proportional to and thus phenomenologically irrelevant. Foreword 11 G U Another firm prediction of GUTs are magnetic monopoles [18, 19]. Each time a simple gauge group is broken to a subgroup with a (1)SU factor there are topologically − nontrivial configurationsQm of theπ/e Higgs field whichMm leadsαU M toU stable monopole solutions of the gauge potential. For instance the breaking of 1 (5) generates a monopole with magnetic charge = 2 and mass = [20]. The central core of a GUT monopole contains the fields of the superheavy gauge bosons which mediate proton decay, so one expects that baryon number can be violated in baryon-monopole scattering. Quite surprisingly it was found [21, 22, 23] that these processes are not suppressed by powers of the unification mass, but have a cross section typical of the strong interactions. Though GUT monopoles are too massive to be produced at accelerators, they could have been produced in the early universe as topological defects arising via the Kibble mechanism [24] during a symmetry breaking phase transition. Experimentally one − − − − tries< to measure their interactions as they pass through matter. The strongest bounds − ÷ − − − on the flux16 < of monopoles2 1 1 come from their interactions with the galactic magnetic field (Φ 10 cm 18sr 29 sec 2) and1 the1 catalysis of proton decay in compact astrophysical objects (Φ 10 cm sr sec ) [11]. Summarizing the model independent predictions of grand unification are proton decay, magnetic monopoles and charge quantizationM (andU /M theW ir∼ deep connection). How- ever once we have a specific model we can do even more. For insta13 nce the huge ratio ÷ between themf unification/mν ∼ and the electroweak scale, 10 , reminds us about the well established hierarchy7 13 among the masses of charged fermions and those of neutrinos, 10 . This analogy hints to a possible connection between GUTs and neutrino masses. The issue of neutrino masses caught the attention of particle physicists since a long time ago. The model independentd way to parametrize them is to consider the SM as an effective field theory by writingY all the possible operators compatible with gauge ν ℓT H C HT ℓ : invariance. Remarkably at the =L 5 level there is only one operator [25] hHi2 v 2 ( ) ( ) (2) Λ v After electroweak symmetry breakingMν =Yν and: neutrinos pick up a Majorana mass term 2L √ m : = atm ∼ (3) Λ The lower bound on the highest neutrino. Y eigenvalue÷ inferred: from ∆ 0 05 eV L ν O tells us that the scale at which the lepton number14 15 is violated is Λ (10 GeV) (4) Actually there are only three renormalizable ultra-violet (UV) completion of the SM which can give rise to the operator in Eq. (2). They go under the name of type- I [26, 27, 28, 29, 30], type-II [31, 32, 33, 34] and type-III [35] seesaw and are respectively Foreword 12 ; ; F ; ; H ; ; F obtainedL by introducing a fermionic singletℓ (1 1 0)H , a scalar triplet (1 3 +1) and a fermionic triplet (1 3 0) . These vector-like fields, whose mass can be identified with Λ , couple at the renormalizable level with and so that the operator in Eq. (2) is generated after integrating them out. Since their mass is not protected by the chiral symmetry it can be super-heavy, thus providing a rationale for the smallness of neutrino masses. Yν L Notice that this is still an effective field theoryYν language and we cannot tell atL this level if neutrinos are light because is small or because ΛYν is large.L It is clear that4 SOwithout a theory that fixes the structure of we don’tSO have much to say about Λ . As an example of a predictive theory which can fix both and Λ we can mention (10) unification. The most prominentYν feature of (10) is that a SM fermion family B L SO plus a right-handedL neutrino fit into a single− 16-dimensional spinorial representation. In turnM thisB−L . readilyMU implies that is correlated to the charged fermion Yukawas. At the same time Λ can be identifiedSO with the generator of (10), and its breaking scale, , is subject to the constraints of gauge coupling unification. Hence we can say that (10) is also a theory of neutrinos, whose self-consistency can be tested against complementary observablesSO such as the proton lifetime and neutrino masses. The subject of this thesis will be mainly (10) unification. In the arduousSO attempt of describing the state of the art it is crucial to understand what has been done so far. In this respect we are facilitated by Fig. 2, which shows the number of (10) papers per year from 1974 to 2010. •By looking÷ at this plot it is possible to reconstruct the following historical phases: τ 1974 1986: Golden age of grand unification.p ∼ These are the years of the foundation in which the fundamentalτp & aspects of the31 theory are worked out. The first estimate of the proton lifetime30 yields 10 yr [37], amazingly close to the experimental bound 10 yr [38]. Hence the great hope that proton decay is • behind÷ the corner. 1987 1990: Great depression. Neither proton decay nor magnetic monopoles are observed so far. Emblematically the last workshop on grand unification is & • held in 1989 [39]. 1991: SUSY-GUTs. The new data of the Large Electron-Positron collider (LEP) seem to favor low-energy supersymmetry as a candidate for gauge coupling unification. From now on almost all the attention is caught by supersymmetry. L 4 The other possibility is that we mayL probe experimentallyV A the new degrees of freedom at the scale Λ in such a way to reconstruct the theory of neutrino masses. This could be the case for left-right symmetric theories [30, 34] where Λ is the scale of the + interactions. For a recent study of the interplay between LHC signals and neutrinoless double beta decay in the context of left-right scenarios see e.g. [36]. Foreword SOH10L 13

10 yr 1980 1985 1990 1995 2000 2005 2010

Figure 2: Blue: number of papers per year with the keyword " (10)" in the title as a function of the years. Red: subset of papers with the keyword "supersymmetry" either in the title or in the abstract. Source:& inSPIRE. • SO 1998: Neutrino revolution. Starting from 1998 experiments begin to show that atmospheric [40] and solar [41] neutrinos change ﬂavor. (10) comes back with & • a rationale for the origin of the sub-eV neutrino mass scale. 2010: LHC era. Has supersymmetry something to do with the electroweak scale? The lack of evidence for supersymmetry at the LHC would undermine & τ ÷ • SUSY-GUT scenarios. Back to nonsupersymmetric GUTs? p ∼ 34 35 2019: Next generation of proton decay experiments sensitive to 10 yr [42]. The future of grand uniﬁcation relies heavily on that.

Despite the huge amount of work done so far, the situation does not seem very clear at the moment. Especially from a theoretical point of view no model of grand unification emerged as "the" theory. The reason can be clearly attributed to the lack of experimental evidence on proton decay. In such a situation a good guiding principle in order to discriminate among models and eventually falsify them is given by minimality, where minimality deals interchange- ably with simplicity, tractability and predictivity. It goes without saying that minimality could have nothing to do with our world, but it is anyway the best we can do at the moment. It is enough to say that if one wants to have under control all the aspects of the theory the degree of complexity of some minimal GUT is already at the edge of the tractability. Quite surprisingly after 37 years there is still no consensus on which is the minimal theory. Maybe the reason is also that minimality is not a universal and uniquely defined concept, admitting a number of interpretations. For instance it can be understood as a mere simplicity related to the minimum rank of the gauge group. This was indeed Foreword 14 SU the remarkable observation of Georgi and Glashow: (5) is the unique rank-4 simple group which contains the SM and has complex representations. However nowadays we can say for sure that the Georgi-Glashow model in its5 original formulation is ruled out because it does not unify andSO neutrinos are massive . SU From a more pragmatic point of view one could instead use predictivity as a measure of minimality. This singlesSO out (10) as the best candidate. At variance with (5), the fact that all the SM fermionsSO of one family fit into6 the same representation makes the Yukawa sector of (10) much more constrained . Actually, if we stick to the (10) case, minimality is closely related toSO the complexity of the symmetry breaking sector. Usually this is the most challenging and arbitrary aspect of grand unified models. While the SM matter nicely fit in three (10) spinorial families, this synthetic feature has no counterpart in the Higgs sector where higher- dimensional representations are usually needed in order to spontaneously break the enhanced gauge symmetry down to the SM. Establishing the minimal Higgs content needed for the GUT breaking is a7 basic question which has been addressed since the early days of the GUT program . Let us stress that the quest for the simplest Higgs sector is driven not only by aesthetic criteria but it is also a phenomenologically relevant issue related to the tractability and the predictivity of the models. Indeed, the details of the symmetry breaking pattern, sometimes overlooked in the phenomenological analysis, give further constraints on the low-energy observables such as the proton decay and the effective SM flavor structure. For instance in order to assess quantitatively the constraints imposed by gauge coupling unification on the mass of the lepto-quarks resposible for proton decay it is crucial to have the scalar spectrum under control. Even in that case some degree of arbitrariness can still persist due to the fact that the spectrum can never be fixed completely but lives on a manifold defined by the vacuum conditions. This also means that if we aim to a falsifiable8 (predictive) GUT scenario, better we start by considering a minimal Higgs sector . 5 Moved by this double issue of the Georgi-Glashow model, two minimal extensions which can cure at the6 same time both unification and neutrino masses have been recently proposed [43, 44]. Notice that here we do not have in mind flavor symmetries, indeed the GUT symmetry itself already constrains the flavor structure just because some particles live together in the same multiplet. Certainly one could improve the predictivity by adding additional ingredients like local/global/continuous/discrete symmetries on top of the GUT symmetry. However, though there is nothing wrong with that, we feel that it would be a no-ending process based on assumptions which are difficult to disentangle from the unification idea. That is why we prefer to stick as much as possible to the gauge principle without further7 ingredients. Remarkably the general patterns of symmetry breaking in gauge theories with orthogonal and unitary groups were already analyzed in 1973/1974 by Li [45], contemporarily with the work of Georgi and8 Glashow. SO As an example of the importance of taking into account the vacuum dynamics we can mention the minimal supersymmetric model based on (10) [46, 47, 48]. In that case the precise calculation of the mass spectrum [49, 50, 51] was crucial in order to obtain a detailed fitting of fermion mass parameters Foreword 15 SO The work done in this thesis can be understood as a general reappraisal of the issue of symmetry breaking in (10) GUTs, both in their ordinary and supersymmetric realizations. We can already anticipateSO that, before considering9 any symmetry breaking dynamics, at least two Higgs representations are required by the group theory in order to SU achieve• H a full breakingH of (10) to the SM: SU U • 16H or 126H : they reduceH the rank but leave an (5) little group unbroken.⊗ SU 45 or 54 or 210 : they admit for little groups different from (5) (1), yielding the SM when intersectedH H with (5). H H H While theH choice between 16 or 126 is a model dependent issue related to the details of the Yukawa sector, the simplest option among 45 , 54 and 210 is given by the SU U adjoint 45 . ⊗ H H However,H since the early 80’s, it has been observed that the vacuum dynamics aligns the adjoint along an (5) (1) direction, making the choice of 16 (or 126 ) and 45 alone not phenomenologically viable.SU In the nonsupersymmetric case the alignment is only approximate [56, 57, 58, 59], but it is such to clash with unification constraints which do not allow for any (5)-like intermediate stage, while in the supersymmetric limit the alignment is exact due to F-flatness [60, 61, 62],H thus never landing to a supersymmetric SM vacuum. The focus of the thesis consists in the critical reexamination of these two longstanding no-go for the settings with a 45 driving the GUT breaking. SO Let us first consider the nonsupersymmetric case. We start by reconsidering the issue of gauge coupling unification in ordinary (10) scenarios with up to two intermediate mass scales, a needed preliminary step before entering the details of a specific Umodel. After complementing the existing studies in several aspects, as the inclusion of the (1) gauge mixing renormalization at the one- and two-loop level and the reassessment of the two-loop beta coefficients, a peculiar symmetry breaking pattern with just the adjoint representation governing the first stage of the GUT breaking emerges as a potentially viable scenarioSO [63], contrary to what claimed in the literatureH [64]. HThis brings us to reexamine the vacuum of the minimal conceivable Higgs potential responsibleH ⊕ H for the (10) breaking to the SM, containing an adjoint 45 plus a spinor SU U SU 16 . As already remarked, a⊗ series of studies inh theH i early 80’s [56, 57, 58, 59] ofh theH i 45 16 model indicated that the only intermediate stages allowed by the scalar sector dynamics were (5) (1)SO for leadingÏ 45 or (5) for dominant 16 H. and9 show a tension between unification constraints and neutrino masses [52, 53]. It shouldF be mentionedF that a one-step (10) SM breaking can be achieved via only one 144 irreducible Higgs representation [54]. However, such a setting requires an extended matter sector, including 45 and 120 multiplets, in order to accommodate realistic fermion masses [55]. Foreword 16 SU SO Since an intermediate (5)-symmetric stage is phenomenologically not allowed, this observation excluded the simplest (10) Higgs sector from realistic consideration. One of the main results of this thesis is the observation that this no-go "theorem" SU SU U is actually an artifact of the tree-level potential and, as we haveC ⊗ shownL ⊗ in Ref.R [65] SU SU SU U (see alsoC ⊗ Ref.L [66]⊗ forR a⊗ briefB−L overview), the minimization of the one-loop effective potential opens in a natural way also the intermediate stageSOs (4) (2) (1) and (3) h H i(2) (2) (1) , which are the options favoured by gauge unification. This result is quite general, since it applies whenever the (10) breaking is triggered bySO the 45 (while other Higgs representations control the intermediate and weak scale stages) and brings back from oblivion the simplestH scenario⊕ H ⊕of nonsupersymmetricH H (10) unification. It is thenSO natural to consider the Higgs system 10 16 45 (where the 10 is needed to give mass to the SM fermions at the renormalizable level) as the potentially minimal (10) theory, as advocated long ago by Witten [67]. However, apart from issues related to fermion mixings, the mainB obstacleL with such a model is given by M −α /π M /M M M neutrino masses. They can be generatedR radiatively∼ U atB−L the twUo-loop level, butB−L turn≪ outU to be too heavy. The reasonMR being that the breaking2 2 is communicated to right- ÷ handed neutrinos at the effective level ( ) and since by13 unification14 constraints, undershoots by several orders of magnitude the value 10 GeV naturally suggested by the type-I seesaw. At these point one can consider two possible routes. Sticking toH the request ofH Higgs representations with dimensions up to the adjoint one can invoke TeV scale supersymmetry, or we can relax this requirement and exchange the 16 with the 126 in the nonsupersymmetricMB−L case. MU In the formerM caseR the gauge running within the minimalF superF HsymmetricH /MP SM (MSSM) prefers in the proximity of so that one can naturally reproduce the desired range for , emerging from the effective operator 16 16 16 16SO . Motivated by this argument, we investigate under which conditions an Higgs sector containing only representations up to the adjoint allows supersymmetric (10) GUTs to breakSO spontaneously to the SM. Actually it is well known [60, 61, 62] that the relevant superpotential does not support, at the renormalizable level, a supersymmetric breaking of the (10) gauge group to the SM. Though the issue can be addressed by giving up renormalizability [61, 62], this option may be rather problematic due to the active role of Planck induced operators in the breaking of the gauge symmetry. They introduce an hierarchy in the mass spectrum at the GUT scale which may be an issue for gauge SO U unification, proton decay and neutrino masses. ⊗ In this respectH we pointedH ⊕ out [68]H that the minimal Higgs scenario that allows for a SO U renormalizable breaking to the SM⊗ is obtained considering flipped (10) (1) with one adjoint 45 and two 16 16 Higgs representations. Within the extended (10) (1) gauge algebra one finds in general three in- equivalent embeddings of the SM hypercharge. In addition to the two solutions with Foreword SU SU U SO 17 SU ⊗ SO SO U the hypercharge stretching over the (5) or the (5) (1) subgroups⊗ of (10) (respectively dubbed as the “standard” and “flipped” (5)U embeddings [69, 70]), there is a third, “flipped” (10) [71, 72, 73], solution inherent to the SU(10) (1) case, with a non-trivial projectionSO of the SM hypercharge ontoSO the (1) factor. Whilst the difference between the standard and theSO flipped (5) embedding is se- mantical from the (10) point of view, the flipped SO (10) case is qualitatively different. In particular, the symmetry-breaking “power” of the (10) spinor and adjoint representations is boosted with respect to the standard (10) case, increasing the number of SM singlet fields that may acquire non-vanishing vacuum expectation values (VEVs). This is at the root of the possibility of implementing the gauge symmetry breaking by E SO U means of a simple renormalizable Higgs sector. ⊗ The model is rather peculiar in the flavor sector and can be naturally embedded 6 in a perturbative grand unified scenario above the flipped (10) (1)H partial- unificationH scale. On the otherMB−L hand,/MU sticking to the nonsupersymmetricd case with a 126 in place of a 16MR, neutrinoMB−L masses are generated at the renormalizable level. This lifts the ∼ SO problematic suppression factor inherentH ⊕ toH ⊕ the H= 5 effective mass and yields , that might be, at least in principle, acceptable. As a matter of fact a nonsupersymmetric (10) model including 10 45 126 in the Higgs sector has all the ingredients to be the minimal realistic version of the theory. This option at the timeH ⊕ of writingH the thesis is subject of ongoing research [74]. Some preliminary results are reported in the last part of the thesis. We have performedMB−L the minimization of the 45 126 potential and checked that the vacuum constraints allow for threshold corrections leading to phenomenologically reasonable values of . If the model turned out to lead to a realistic fermionic spectrum it would be important then to perform an accurate estimate of the proton decay branching ratios.SO The outline of the thesis is the following:SU the first Chapter is an introduction to the field of grand unification. The emphasis is put on the construction of (10) starting from theSO SM and passing through (5) and the left-right symmetric groups. The second Chapter is devoted to the issue of gauge couplingsU unification in nonsupersymmetric (10). A set of tools for a general two-loop analysis of gauge coupling unifi- SO cation, like for instance the systematizationH ⊕ H of the (1) mixing running and matching, is also collected. Then in the third Chapter we consider the simplest and paradigmatic (10) Higgs sector made by 45 16 . After reviewing the old tree level no-go argument we show, by means of an explicit calculation, that the effective potential al- SO lowsH ⊕ forH those⊕ H patterns which were accidentally excluded at tree level. In the fourth Chapter we undertake the analysis of the similar no-go present in supersymmetry with 45 16 16 in the Higgs sector. The flipped (10) embedding of the hypercharge is proposed as a way out in order to obtain a renormalizable breaking with only representations up to the adjoint. We conclude with an Outlook in which we suggest the possible lines of development of the ideas proposed in this thesis. The case is made Foreword 18 SO for the hunting of the minimal realistic nonsupersymmetric (10) unification. Much of the technical details are deferred in a set of Appendices. Chapter 1

From the standard model to O S

SO (10)

InSU this chapter we give the physical foundations of (10) as a grand unified group, starting from the SM and browsing in a constructive way through the Georgi-Glashow (5) [1] and the left-right symmetric groups such as the Pati-Salam one [2]. This will offer us the opportunity to introduce the fundamental concepts of GUTs, as charge quantization, gauge unification, proton decay and the connection with neutrino masses in a simplifiedSO and pedagogical way.

The (10) gauge groupSO as a candidate forSU the unification of the elementary interactions was proposed long ago by Georgi [75] and Fritzsch and Minkowski [76]. The main advantage of (10) with respectSO to (5) grand unification is that all the known SM fermions plus three right handed neutrinos fit into three copies of the 16-dimensional spinorial representation of (10).SU In recent years the field received an extra boost due to the discovery of non-zero neutrino massSOes in the sub-eV re- B L gion. Indeed, while in the SM (and similarly− in (5)) there is no rationale for the origin of the extremely small neutrino mass scale, the appeal of (10) consists in the predictive connection between the local16 breaking scale (constrained by gauge coupling unification somewhat below 10 GeV) and neutrino massesSO around 25 orders of magnitude below. Through the implementation of some variant of the seesaw mechanism [26, 27, 28, 29, 30, 31, 32, 33, 34] the inner structure of (10) and its breakingSO makes very natural the appearance of such a small neutrino mass scale. This striking connection with neutrino masses is one of the strongest motivations behind (10) andV it canA be traced back to the left-right symmetric theories [2, 77, 78] which provide a direct connection of the smallness of neutrino masses with the non-observation of the + interactions [30, 34]. SO

Chapter 1. From the standard model to 1.120 The standard model chiral structure (10)

SU C ⊗ U Q The representations of the unbroken gauge symmetry of the world, namely (3) (1) , are real. In other words, for each colored fermion field1 of a given electric charge we have a fermiong field of opposite colorG and charge . If notD g so we would ∗ observe for instance a massless charged fermion field and thisD isg not the case. More formally, being an element of a group , a representation ( ) is said to be real (pseudo-real) if it is equalSD g toS− its conjugateD∗ g representg ationG; ( ) up to a similarity transformation, namely ∈ S 1 ( ) = ( ) forall (1.2) whit symmetric (antisymmetric).S A complex representation is neither real norTa pseudo-real. G D g igaTa It’s easy to prove that must be either symmetric or antisymmetric. Suppose ST S− T∗ : generates a real (pseudo-real) irreduciblea unitary− a representation of , ( ) = exp , 1 so that Ta − T = − T T T (1.3) STaS T S T S Ta ; Because the are hermitian,− wea can write a − 1 1 = or− T ( − ) T = (1.4) Ta S STaS S which implies 1 1 − T =Ta (; S )S : (1.5) or equivalently 1 =0 (1.6) But if a matrix commutesS with− ST allλI the generatorsST ofλS: an irreducible representation, Schur’s Lemma tells us that it is a multiple of the identity, and thus 1 ψL iσω 1 = ψL or = ψL Ï e ψL (1.7) µν i T Asσω is usual ≡ω inσµν grandσµν ≡ unificationγµ; γν we useγµ the; γν Weylg notationµν in which all fermion fields areψL Cψ left-L T { } − T handedC (LH) four-componentσµνC spinors.Cσµν Given a fieldC transformingγµC γµ as under the Lorentz 2− − group (γ , and = 21 ) an invariant mass term is given by σ where is such that = γ or (up to a sign); γi = i. Using; the following representation for the matrices σi 0 − 0 1 0 − † T σi = C = C iγ γ C C C C(1.1) 1 0 0 − iθ − − U ψL e ψ1L 2 0 Ï where are the Pauli matrices, an expression for reads = ψL , with = = = . ′T ′ Notice that the mass termψL isCψ notL invariant under the (1) transformation ψL ψLand in order to ′ avoid the breaking of any abelianψL quantum number carried by (such as lepton numberψL or electric charge) we can construct where for every additive quantum number andψR have opposite ′T ψ ψ C ψR ψ ψL charges.R ≡ L This just means that if is associated with a certain fundamental particle, is associatedR with its antiparticle. In order to recast a more familiar notation let us define a field by the equation . In therms of the right-handed (RH) spinor , the mass term can be rewritten as . 1.2. The Georgi-Glashow route λ 21 λ S ± 2 By transposing twice we get back to where we started and thus we must have = 1 and so = 1, i.e. must beψL either symmetric or antisymmetric. ψ D g ψ The relevanceL Ï of thisL fact for the SM is encoded in the following observation: given a left-handed fermion field transforming under some representation, reducible or T irreducible,D g ( ) , oneψL canCSψ constructL C a gauge invariant mass term only if the representation is real. Indeed, it is easy to verify (by using Eq. (1.2) and the unitarity of SU( )) that the mass term , where denotes the Dirac chargeS conjugation matrix, is invariant. Notice that if the representation were pseudo-real2 (e.g. a doublet of (2)) the mass term vanishes because of the antisymmetry of . SU U The SM is built in such a wayL ⊗ that thereY are no bare mass terms and all the masses stem from the Higgs mechanism. Its representations are said to be chiral because they are chargedG under the (2) (1) chiral symmetry in such a way thatG fermions SU SU U SU U are massless as long as the chiralC symmetry⊗ L ⊗ is preserved.Y Ï AC co⊗mplexQ representation of a group may of course become real when restricted to a subgroup of . This is exactly what happens in the (3) (2) (1) (3) (1) case. When looking for a unified UV completion of the SM we would like to keep this feature. Otherwise we should also explain why, according to the Georgi’s survival hypothesis [79], all the fermions do not acquire a super-heavy bare mass of the order of the scale at which the unified gauge symmetry is broken. 1.2 The Georgi-Glashow route

SU SU U The bottom line of the last sectionC ⊗ was thatL a⊗ realisticY grand unified theory is such that the LH fermions are embedded in a complex representation of the unified group (in SU SU U particular complex under (3) (2) (1) ). If we furtherC require⊗ minimalityL ⊗ Y (i.e. rank 4 as inSU the SM) one reaches the remarkable conclusion [1] that the only simple group with complex representations (which containsSU (3) (2) (1) i i ;:::; SU C SU L as a subgroup) is (5). ⊗ Let us consider the fundamental representation of (5) andSU denoteC it as a 5- dimensional vector 5 ( = 1 SU5).L It is usual to embed (3) (2) in such a way that the first three components of 5 transform as a triplet of (3) and the last ; ; : two components as a doublet of (2) ⊕ 5=(3 1) (1 2) (1.8) q ; ; ℓ ; ; uc ; ; dc ; ; ec ; ; : In the∼ SM we have∼ 15 Weyl− fermions∼ per family− with∼ quantum numb∼ ers 1 1 2 1 T 6C C 2 3 3 (3 2 + ) − (1 2 ) (3 1 ) (3 1 + ) (1 1 +1) (1.9) 2 The relation = and the anticommuting property of the fermion fields must be also taken into account. SO

Chapter 1. From the standard model to 22 SU SU(10)

How to embed these into (5)? One would be tempted to try with a 15 of (5). A S ; Actually from the tensor product ⊗ ⊕

A S ⊗ ⊕ 5 5=10 15 (1.10) ; ⊕ and the fact that 3⊕ 3 = 3 6 one concludes that some of the known quarks should SU C SU L U Y belong to color sextects, which is⊗ not the case.⊗ So the next step is to try with 5 10 or better with 5 10 since there is no (3 1) in the set of ﬁelds in Eq. (1.9). The ; ; ; ; ; decomposition of 5 under (3) (2) ⊕ (1) −is simply 1 1 5 = (3 1 + 3 ) (1 2 2) SU(1.11) Y dc Y ℓ Y dc Y ℓ where we have exploited the fact that− the hypercharge is a traceless generator of (5), which implies the condition1 3 ( )+21 ( ) = 0. So, up to a normalization factor, one 3 2 A ; ; ; ; ; ; : may choose ( ) = and⊗ ( ) = . Then− ⊕ from Eqs. (1.10)–(1.11)⊕ we get 2 1 3 6 10 = (5 5) = (3 1 ) (3 2 +⊕) (1 1 +1) (1.12) dc uc uc u d Thus the embedding of a SM fermion family into 5 −10 reads  dc uc uc u d 1c   − c 3c 2 1 1  d ; u0 u u d ;  2 3 − 1 2 c2  e   u 0u u e   3   − 2 − 1 − 3c 3  5 =  ν  10 =  d d 0d e  (1.13)  −   − 1 − 2 − 3 −    0  1 2 3 SU L q u d 0ℓ ν e iσ ℓ ℓ∗ ∼ where we have expressed the (2) doubletsSU as =3 ( ) and = ( ). Notice in 2 particular that the doublet embedded in 5 is . It may be useful to know how the (5) generators act of 5 and 10. From the i † i k k l transformation properties U k ; ij Ui Uj kl ; Ï Ï † U iT T5 T( ) 5 10 10 (1.14)

i i k where = exp and δ= , weTk deduce; that δ theij actionT; ofij the: generators is − { } 5 = 5 10 = 10 SU (1.15)

Alreadyσ at this elementary level we can list a set of important featuresiσ ℓ of ℓ∗(5) which are typical of any GUT. ∼ 3 SU L 2 2 Here is the second Pauli matrix and the symbol " " stands for the fact that and transform in the same way under (2) . 1.2. The Georgi-Glashow route 1.2.1 Charge quantization and anomaly cancellation 23

SU C SU The charges of quarks and leptonsQ area;a;a;b; related. Leta usb write; the most general electric charge generator compatible with the (3) invariance− − and the (5) embedding Q = diag ( 3 ) (1.16) Q dc a Q e b Q ν a b where Tr = 0. Then by applying− Eq. (1.15)− we find Q uc a Q u a b Q d a b Q ec a; ( ) = ( ) = − ( )=3 + − (1.17) ( )=2 ( ) = + ( ) = (2 + ) ( ) = 3 Q(1.18)ν so that apart for a global normalization factor the charges do depend just on one parameter,4 whichQ ec mustQ bee fixedQ byu some extraQ uc assumption.Q d Let’Qs saydc web; require ( ) = 0 , that readily implies− − − 3 3 b ( ) = ( ) = 2 ( ) = 2 ( ) = 3 ( )=3 ( ) = (1.19) i.e. the electric charge of the SM fermions is a multiple of 2 . SU Let us consider now the issue of anomalies. We already know that in the SM SU all the gauge anomalies vanish. This property is preserved in (5) since 5 and 10⊃ SU U A SU U A U B have equal⊗ and⊃ opposite⊗ anomalies,⊗ so that the theory is still anomaly free. In order to see this explicitly let us decompose; 5 and; 10 under; the; branching chain (5) (4) (1) (3)⊕ − (1) (1)⊕ − ⊕ − − ; ; ; ; ; 5 = 1(4) ⊕4( −1) = 1(4 0) ⊕1( 1 −3) ⊕3( −1 −1) ⊕ − − (1.20) U R 10 = 4(3) 6( 2)R = 1(3 3) 3(3 1) 3( 2 2) 3( 2 2) (1.21)A where the (1) charges are givena upb to ac normalizationabc factor. The anomaly ( ) TR;TR TR R d ; relative to a representation is{ defined} by A dabc Tr = ( ) (1.22) R R R R R R ; where is aA completely⊕ symmetricA A tensor. Then, givenA the propertie−A s 1 1 1 2 SU SU ( ) = ( ) + ( ) and ( ) = ( ) (1.23) SU SU ; SU SU SU SU ; it is enoughA to computeA the anomalyA of theA (3) subalgebraA of A(5), (3) (3) (3) (3) (3) (3) (5) = (3) A ⊕ (10) = (3) + (3) + (3) (1.24) in order to conclude that (5 10) = 0. We close this section by noticing that anomaly cancellation and charge quantization are closely related. Actually it is not a chance that in the SM anomaly cancellation implies charge quantization, after taking into account the gauge invariance of the Yukawa couplings4 [6, 7, 8, 9, 10]. SU L H H U Q Q H ⊂ That ish neededi in order to give mass to the SM fermions with the Higgs mechanism. The simplest possibility is given by using an (2) doublet 5 (cf. Sect. 1.2.6) and in order to preserve (1) it must be ( )=0. SO

Chapter 1. From the standard model to 1.2.224 Gauge coupling uniﬁcation (10)

MU SU ′ g g g SU C SU L U Y ⊗ ⊗ ′ At someg grandU unification mass scale theg relevantg g symmetry is (5) and the 3 2 , , coupling constants of (3) (2) (1) merge into one single gauge 3 2 coupling . The rather different values for , , at low-energy are then due to ′ renormalization effects.g g Before considering the running of the gauge couplings we need to fix the relative 2 g T g ′Y: normalization between and , which enter the weak interactions 2 3 + Y (1.25) ζ ; We define T2 Y ζ− / Y T Tr 2 = 3 (1.26) ≡ Tr′ 1 2 g Y g Y 1 3 so that is normalized as . In a unified theory based on a simple group, 1g 1 ζg ′ : the coupling which unifies is then ( ≡= ) 1 SUp (1.27)

Evaluating the normalization overζ a 5 of (5)− one ﬁnds; 2 2 1 1 3 − 2 3 2 +2 2 5 = 1 1 = (1.28) 2 + 2 3

gU g MU g MU g MU : and thus one obtains the tree≡ level matching condition

log10HΜGeVL 5 10 15 18

U Y SU

Figure 1.1: One-loop running of the SM gauge couplings assuming the (1) embedding into (5).

As weMU can see, the gauge couplings do not unify in the minimal framework, although a small perturbation may suffice to restore unification. In particular, thresholds effects at the scale5 (or below) may do the job, however depending on the details of the UV completion . ÷ By now Fig. 1.1 remains one of the most solid hints in favor of the grand unification idea. Indeed, being the gauge coupling evolution sensitive to the log15 of18 the scale, it is intriguing that they almost unify in a relatively narrow window, 10 GeV, which is still allowed by the experimental lower bound on the proton lifetime and a consistent 1.2.3effective Symmetry quantum field breaking theory description without gravity.

H H ⊕SU SU C SU L U Y SU C U Q The Higgs sector of the Georgi-Glashow⊗ ⊗ model spans over the reducible⊗ 5 24 representation. These two ﬁelds are minimally needed in order to break the (5) H gauge symmetryh downi to (3) (2) (1) and further to (3) (1) . Let us concentrate on the ﬁrst stage of the breaking which is controlled by the rank- conserving VEV 24 . The fact thati the adjoint preservesi the rank is easily seen by δ H T ; H ; considering the action of the Cartanh ij generatorsh on thei j adjoint vacuum Cartan 24 = [ 24 ] (1.32) i U† i Ul k : derived from the transformation propertiesj Ï k ofj thel adjoint

H SU h i 24 ( ) 24 (1.33) Since 24 can be diagonalized by an (5) transformation and the Cartan genera- tors5 are diagonalSU by deﬁnition, one concludes that the adjoint preserves the Cartan It turns out that threshold corrections are not enough in order to restore uniﬁcation in the minimal Georgi-Glashow (5) (see e.g. Ref. [81]). SO

Chapter 1. From the standard model to 26 (10)

V H m λ λ ; subalgebra. The scalar potential− is givenH by H H 2 2 2 2 4 1 2 H H (24 ) = Tr24 + Tr24 + Tr24 Ï − (1.34) H h i where just for simplicity we have imposed theSU discrete symmetry 24 24 . The minimization of the potential goes as follows. First of all 24 is transformed into a H h ; h ; h ; h ; h ; real diagonal traceless matrixh byi means of an (5) transformation 1 2 3 4 5 h h h h h 24 = diag( H ) (1.35)

1 2 3 4 5 where + + + + = 0. With 24 in the diagonal form, the scalar potential V H m h λ h λ h : reads − i i 2 i 2 i 2 i 2! i 4 X 1 X 2 X hi (24 ) = + + (1.36)µ

i hi ′ V H V H µ H Since the ’s are− not all independent,P we need to use the lagrangian multiplier in order to account for the constraint = 0. The minimization of the potential ′ (24 ) = (24 ∂V) Tr24H yields m hi λ hj hi λ hi µ : ∂hi −   − 2 j 2 3 (24 ) 1 X 2 = 2 +4   +4 =0 (1.37) hi

λ x λ a m x µ a h : Thus at the minimum all the − ’s satisfy− the same cubic equation j 3 2 j 2 2 1 X 4 + 4 2 = 0 with = (1.38) hi φ φ φ x 1 2 3 This means that the ’s can take at most three diﬀerent values, , and2 , which are the three roots of the cubic equation.φ φ Noteφ that: the absence of the term in the cubic equation implies that n n n 1 φ 2 φ 3 φ H + + =0 h i (1.39) 1 2 3 1 2 3 H φ ;:::;φ ;:::;φ n φ n φ n φ : Let ,h andi the number of times , and appear in 24 , 1 2 3 1 1 2 2 3 3 24H = diag( SU n )SU withn SU n + + =0 (1.40) h i ⊗ ⊗ SU n SU n SU n SU n 1 2 3 Ï ⊗ ⊗ Thus 24 is invariantU under ( ) ( ) ( ) transformations.H This implies h i1 2 3 that the most general form of symmetry breaking is ( ) ( ) ( ) ( ) as well as possible (1) factors (total rank is 4) which leave 24 invariant. To ﬁnd the absolute minimumn φ wen φ haven toφ use the relations φ φ φ

1 1 2 2 3 3 1 2 3 + + = 0 and + + =0 (1.41) 1.2. The Georgi-Glashow route n ; n ; n 27 { } V H 1 2 3 to compare diﬀerent choices of in order to get the one with the smallest (24 ). It turns out (see e.g. Ref. [45]) that for the case of interest there are two possible SU SU SU U SU SU U ; patterns for the symmetryÏ ⊗ breaking⊗ Ï ⊗ λ λ (5) (3) (2) (1) or (5) (4) (1) (1.42) λ > λ > 1 2 depending on the relative magnitudes of the parameters and . In particular for 1 2 H V ; ; ; ; : 0 and 0 the absoluteh i minimum is given− by− the SM vacuum [45] and the adjoint VEV reads 24 = diag(2 2 2 3 3) (1.43)

Then the stabilityλ of theH vacuumλ requiresH > λ > λ h i 2 h i ÍÑ − 2 4 1 2 1 7 2 Tr 24 + Tr 24 0 (1.44) 30 ∂V H h i V m V λ λ and the minimum∂V condition ÍÑ − 2 2 ( 24 ) 1 2 =0 60 +2 (30 +7 ) =0 (1.45) m V : yields λ 2 λ 2 = 1 2 (1.46) 2(30 +7 ) D ∂ ig A ; ; Let us now write the covariantµ derivativeH µ H µ H

Aµ H × 24 = 24 + 24 (1.47) µ † µ Dµ H D H g Aµ; H H ; A where and 24 areh 5 i5 tracelessh i hermitian matrices.h i h Theni from the canonical kinetic term, 2 i i Tr 24 24 =H Trhj δ ; 24 [ 24 ] (1.48) h ij j and the shape of the vacuum 24 = (1.49) where repeated indices are noti summed,µ wej can easilyi extractµ j the gauge bosons mass g Aµ; H H ; A g Aµ A hi hj : matrix from the expressionh i j h i i j i − 2 2 2 i A24µ j [ 24 i ;] =; ( j) ( ;) ( ) MX (1.50)g V i;j ; ; i;j ; 2 2 2 The gauge boson fields ( ) havingEq:=1 2 3 and =4 5 are massive, = 25 , while =1 2 3 and =4 5 are still massless. Notice that the hypercharge− genera- SU /SU C SU L U Y tor commutes with the vacuum in (1.43) and hence the associated⊗ gauge boson⊗ is massless asM well.X The number of massive gauge bosons is then 24 M(8+3+1)U = 12 and their quantum numbers correspond to the coset (5) (3) (2) (1) . Their mass is usually identified with the grand unification scale, . SO

Chapter 1. From the standard model to 1.2.428 Doublet-Triplet splitting (10)

SU C SU L U Y SU C U Q ⊗ ⊗ Ï ⊗ H T The second breaking step, (3) (2) (1); (3) (1) , is driven by a H H 5 where T 5 = SU L H (1.51) H decomposesH into a color triplet and an (2) doublet . The latter plays the same role of the Higgs doublet of the SM. The most general potential containing both 24 V V V V ; ; and 5 can be written as H H H H V H = (24 ) + (5 ) + (24 5 ) (1.52) † † where (24 ) is defined inV Eq.H (1.34), µ H H λ H H ; − 2 2 (5 ) = † 5 5 + 5 5 † (1.53) V H ; H α H H H β H H H : 2 2 and H H (24 5 ) = 5 5 Tr24H + 5 24 5 T Ï − (1.54) Again we have imposed for simplicity the discrete symmetry 24 24 . It is instructive to compute the mass of the doublet and the triplet in the SM vacuum M µ α β V ; M µ α β V : just after the firstH stage− of the breaking T − 2 2 2 2 2 2 M M H = X+ (30 W+9 ) = + (30 +4 ) (1.55) ≫ W Z TheMW gauge hierarchy requires that the doublet , containing the would-be M /M M GoldstoneO bosonsX W eaten∼ by the and the and theH physical Higgs boson, live at the scale. This2 2 is unnatural26 and can be achieved at2 the prize of a fine-tuning ofM oneT part in ( ) 10 in the expression for . If we follow the principle that only the minimal fine-tuning6 needed for the gauge hierarchy is allowed then is automatically kept heavy . This goes under the name of doublet-triplet (DT) splitting. Usually, but not always [84, 85], a light triplet is very dangerous for the proton stability since it can couple to the SM fermions in such a way that baryon7 number is not anymore an accidental global symmetry of the low-energyMX lagrangianMW . A final comment about the radiative stability of the fine-tuning is in order. While su- persymmetry6 helps in stabilizing the hierarchy between and against radiative In some way this is an extension of the Georgi’s survival hypothesis for fermions [79], according to which the particles do not survive to low energies unless a symmetry forbids their large mass terms. This hypothesis is obviously wrong for scalars and must be extended. The extended survival hypothesis (ESH) reads: Higgs scalars (unless protected by some symmetry) acquire the maximum mass ∗ compatible with the pattern of symmetry breakingqqT [82].qℓT In practice this corresponds to the requirement ofT the7 minimal numberU of fine-tuningsB to be imposed onto the scalar potential [83]. Let us consider for instance the invariants and . There’s no way to assign a baryon charge to in such a way that (1) is preserved. 1.2. The Georgi-Glashow route 29 corrections, it does not say much about the origin of this hierarchy. Other mechanisms have to be devised to render the hierarchy natural (for a short discussion of the solutions proposed so far cf. Sect. 4.4.3). In a nonsupersymmetric scenario one needs to compute the mass of the doublet in Eq. (1.55) within a 13-loop accuracy in order to 1.2.5stabilize Proton the hierarchy. decay

SU /SU C SU L U Y The theory predicts that⊗ protons⊗ eventually decay. The most emblematic contribution SUto proton decay is due to the exchange of super-heavy gauge bosons which belong to αβ αi ij the coset (5) (3) (2)ψα; ψi ;(1) . Let usψ denote; ψ ; ψ the matter; representations of (5) as SU C SU L 5 = ( ) 10 = (1.56) where the greek and latin indices runα respectivelyi α i fromα i 1 to 3 ( (3) space) and 1 to Xβ ;Xj ;Xα Xi ;Xi ;Xα ; 2 ( (2) space). Analogously the adjoint 24− can be represented as 3 2 ; ; 24; = (1.57) ; ; ⊕ ⊕ Xα from which⊕ we can readily recognize the gauge bosons associai ted to the SM unbroken generatorsα ((8 1) c(3 1) (1 1))βi and theβα two super-heavyc ij leptoquarkαj c gauge bosons Xi ψα ψi d ν;e ; ψ ψ d;u u ; ψ ψ e u;d : ((3 2) (3 2)).Ï Let usÏ consider nowÏ the gauge actionÏ of onÏ the matterÏ ﬁelds α Xi : ( ) ( ) ( ) (1.58) ud ucec ; Thus diagrams involving the exchange ofÏ a boson generate processes like (1.59) p π e whoseÏ amplitude is proportional to the gauge boson propagator. After dressing the operator0 + with a spectator quark u, we can have for instance the low-energy process αU mp , whose decay rate can bep estimatedπ e by simple: dimensional analysis Ï ∼ M2 X 5 0 + 4 − τ p π e > : Γ( ) αU (1.60) → × 0 + 33 1

Using ( ) 8 2 10 MyearsX > [11]: we extract (for = 40) the naive lower bound on the super-heavy gauge boson mass× 15 2 3 10 GeV (1.61) B L p π e which points directly− to the grand uniﬁcation scale extrapoÏ lated by the gauge running (see e.g. Fig. 1.1). 0 + Notice that Bis conservedL in the process . This selection rule is B L Xα −/ a general feature− ofi the gauge induced proton decay and can be traced back to the presence of a global accidental symmetry in the transitions of Eq. (1.58) after assigning ( )=2 3. SO

Chapter 1. From the standard model to 1.2.630 Yukawa sector and neutrino masses (10) SU

8 ∗ The (5) Yukawa lagrangianY F Y canF be written schematicallyF Y F H as ; L H 5 1 5 10 = 5 10 5 + 10 10 5 + h.c. SU (1.62) 8 5 where is the 5-index Levi-Civita tensor. After denoting the (5) representations dc uc q T synthetically as ; F ℓ F qT ec H H −3 5 = 2 10 = 2 iσ 5 = (1.63)

3 2 2 where is the 3-index Levi-Civitac tensor and ∗ = , we project Eq. (1.62) over the ∗ c T u q T c ∗ c ∗ SM components.F Y F Thisd yieldsℓ d Y qH ℓY e H ; H qT ec H∗ Ï −3 5 2 5 5 2 5 10 5 = c T + (1.64) F Y F H u Y Y qH : Ï 1 5 10 SU 1 10 10 10 10 5 L + (1.65) ℓY ecH∗8 ecY T ℓH∗ 2 After rearranging the order of the (2) doublet and singlet ﬁelds in the second term 5 Y 5 Y T Y Y T ; of Eq. (1.64), i.e. = d e , one gets u u

= and = (1.66) mb MU mτ MU whichms MU showsm aµ M deepU connectionmd MU betweenme M flavorU and the GUT symmetry (which is not related to a flavor symmetry). The first relation in Eq. (1.66) predicts ( ) = ( ), mb MU mτ MU ( ) = ( ) and ( ) = ( ) at the GUT scale. So in order− to test this relation one has to run the SM fermion masses starting from their low-energy values. While ( )m =d/me ( )m iss/m obtainedµ in the MSSM with a typical 20 30% uncertainty [86], the other two relations are evidently wrong. By exploiting the fact that the ratio between and is essentially independent of renormalization m /m m /m ; effects [87], we get the scale free relationd s e µ

= (1.67) md me H SU h i which isSU oﬀ by one order of magnitude.

Notice that = comes∗ from theαx fact thatmn the fundamentaly ∗ β 5 breaks (5) F Y F H F m Cxy Y F αβn H F Y F H downαβγδ to x (4) whichmn remainsy an≡ accidental symmetry of the Yukawa sector. So one≡ 8 F αβm Cxy Y F γδn H α;β;γ;δ; m; n x;y SU 5 5 5 10 More precisely 5 10 5 5 ( ) (10 ) (5 ) and 10 10 5 10 (10 ) ( ) (10 ) (5 ) , where ( ), ( ) and ( ) are respectively (5), family and Lorentz indices. 1.2. The Georgi-Glashow route 31

SU H expects that considering higher dimensional representations makes it possible to further break the remnant (4). This∗ is indeed what happens: by introducing a 45 which couples to the fermionsF inF theH followingF F wayH [88]

Yd Ye H H 5 10 −45 + 10 10 45 + h.c. (1.68) The first operator leads to = 3 , so that if both 5 and 45 are present more ∗ freedom is available to fit all fermionF masses.F H AlternativelH ; y one can built an effective coupling [89] h i 1 H 45 5 10 ( 24 5 ) (1.69) MP b τ − Λ which mimics the behavior of the 45 . If we take the cut-off to be the planckMd scale Me , this nicely keeps unification while corrects the relations among the first two families. However in both cases we loose predictivity since we are just fitting and U G G F in the extended Yukawa structure. − G FinallyF what aboutG H neutrinos? It turnsH out [90] that the Georgi-Glashow model3 h i 5 has an accidental1 global (1)2 symmetryG with the charge assignmentSU (5 ) = , (10 ) = + 5 and (5 ) = +G 5 . TheY VEV 5 breaks this global symmetry but leaves invariant a linear combinationG Y of4 and a Cartan generator of (5). It easyB toL see that 5 − any linear combination of 4+ , Q, and any color generators is left invariant. The B L 5 extra conserved charge −+ when acting on the fermion fields is just . Thus neutrinos cannot acquire neither a Dirac (because of the field content) nor a Majorana (because of the global symmetry) mass term and they remainU exactlyG massless even at the quantum level. Going at the non-renormalizable level we can break the accidental (1) symmetry. For instance global charges are expected to be violated by gravity and the simplest effective operator one can think of is [91]F F H H : MP 1 5 5 5 5 mν MW(1.70)/MP − ∼O ∼ 2 However5 its contribution to neutrino masses is too much suppressed ( ( ) 10 eV). Thus we have to extendSU the field content of the theoryF in order to generate phenomenologicallyF viableF H neutrino masses.F Actually, the possibilities are many. Minimally one may add an (5) singletSU fermion field 1 . Then, through its renormalizable couplingMP 5 1 5 , one integrates 1 out and generates an operator similar to that in Eq. (1.70), but suppressed by the (5)-singlet mass term whichU G can be taken well below . H A slightly different approach could be breaking the accidental (1) symmetry by adding additional scalar representations. Let us take for instance a 10 and consider f F F H M H H H : then the new couplings [90]L ⊃ 10 5 5 10 + 10 10 5 (1.71) SO

Chapter 1. From the standard model to 32 (10) G F G H G H − U G B L f 3 2 − SinceM (5 ) = 5 and (5 ) = + 5 there’s no way to assign a -charge to 10 in order to preserve (1) . Thus we expect that loops containing the breaking sources and can generate neutrino masses. So what is wrong with the two approaches above? In principle nothing. But maybe we should try to do more than getting out what weSU put in. Indeed we are just solving the issue of neutrino masses "ad hoc", without correlations9 to other phenomena. In addition we do not improve uniﬁcation of minimal (5) . Guided by this double issue of the Georgi-Glashow model, two minimal extensions which can cureH at the; H same; timeH both; H neutrino masses; H and uniﬁcation have been • ⊕ ⊕ recently proposed ; H Add a 15 = (1 3) (6 1) (3 2) [43]. Here (1 3) is an Higgs triplet responsible for type-II seesaw. The model predicts generically light leptoquarks (3 2) F ; F ; F ; F ; F ; F ; F ; F • and fast proton decay⊕ [92].⊕ ⊕ ⊕

Add a 24 = (1 1) (1 3); F(8 1) (3 2) (3 2) [44]. Here (1 1) and (1 3) are fields responsible respectively for type-I and type-III seesaw. The model predicts a light fermionSU triplet (1 3) and fast proton decay [93]. Another well motivated and studied extension of the Georgi-Glashow model is given by supersymmetricR (5) [94]. In this case the supersymmetrization of the spectrum is enough in order to fix both unification and neutrino masses. Indeed, if we do not impose by hand -parity conservation Majorana neutrino masses are automatically 1.3generated The by lepton Pati-Salam number violating route interactions [95].

V A In the SM there is an intrinsic− lack of left-right symmetry without any explanation of the phenomenological facts that neutrino masses are very small and the weak interactions are predominantlyu u u . The situationν can be schematicallydc dc dc depicteddc inec the following q ℓ way d d d e uc uc uc uc 1 2 3 1 2 3 c = (c ) c q= ; 1 ; 2 ℓ 3 ; ; = d ; ; u 1 2; ; 3 e ; (1.72); − = ( −) ? SU C SU L U Y ⊗ 1 ⊗ 1 1 2 where = (3 2 + 6 ), = (1 2 2), = (3 1 + 3 ), = (3 1 3) and = (1 1 +1) underd (3) (2) (1) . Considering the SM as an eﬀectiveY theory, neutrino masses can be generated by a ν ℓT H C HT ℓ ; = 5 operator [25] of the type L 2 2 ( ) ( ) (1.73)H 9 Λ An analysis of the thresholds corrections in the Georgi-Glashow model with the addition of the 10 indicates that uniﬁcation cannot be restored. 1.3. The Pati-Salam route iσ C 33 T H v Mνν Cν 2 h i 2 where = and is the charge-conjugation matrix. After electroweak symmetry v breaking, = , neutrinos pick upM aν MajoranaYν : mass term with 2L

= √ matm :(1.74) Λ ∼

The lower bound on the highest neutrino eigenvalue÷ inferred from ∆ 0 05 eV L . Yν : tells us that the scale at which the leptonO number is violated is 14 15 Y Λ (10 GeV) ν (1.75) L σi ab σi cd δadδcb δabδcd Notice that without a theory which fixes the structure of we don’t− have much to say about Λ . Actually, by exploiting the Fierz identity ( ) ( ) =2 , one finds that T T T T T T the operatorℓ H inC Eq.H (1.73) ℓ canℓ beC equivalentlyσiℓ H σiH written inℓ thrσieeH C differentH σiℓ ways: − 2 2 1 2 2 2 2 ( ) ( ) = ( )( ) = ( ) ( ) (1.76) 2 Each operator in Eq. (1.76) hints to a different renormalizable UV completion of the SM. Indeed one can think those effective operators as the result of the integration of ℓT H Cνc ℓT C σ ℓ ℓT σ H CT ; an heavy state with a renormalizable couplingi ofi the type i i

c 2 2 2 ν i Ti Y Y ( ) ( )∆c ( ∗ ) (1.77) Y ν i Ti ⊕ i where , ∆ and are a fermionic singlet ( = 0), a scalar triplet ( = +1) and a fermionicL triplet ( = 0). Notice that being , ∆ ∆ and vector-like states their mass is not protected by the electroweak symmetry and it can be identiﬁed with the scale Λ , thus providing a rationale for the smallness of neutrino masses. This goes under the name of seesaw mechanism and the three options in Eq. (1.77) are classiﬁed 1.3.1respectively Left-Right as type-I [26,symmetry 27, 28, 29, 30], type-II [31, 32, 33, 34] and type-III [35] seesaw.

c Guidedν by the previous discussion on the renormalizable origin of neutrino masses, it is then very natural to to ﬁll the gap in the SM by introducing a SM-singlet fermion ﬁeld . In such a way the spectrum looks more "symmetric" and one can imagine that at higher energies10 the left-right symmetry is restored, in the sense that left and right chirality fermions are assumed to play an identical role prior to some kind of spontaneous10 symmetry breaking. ψ ψcT C ψc Cγ ψ∗ As alreadyR ≡ L stressed we work inL a≡ formalismR in which all the fermions are left-handed four components Weyl spinors. The right chirality components are obtained by means of charge conjugation, 0 namely or equivalently . SO

Chapter 1. From the standard model to 34 (10) SU C SU L SU R ⊗ ⊗ ⊗ U B L Z Z SU L − ⊗ ↔ SU TheR smallest gauge group that implement this idea is (3) (2) (2) 2 2 (1) [2, 77, 78], where is a discrete symmetry which exchange (2) u u u ν dc dc dc ec q(2) . The field contentℓ of the theoryqc can be schematically depicteℓc d as d d d e uc uc uc νc 1 2 3 − 1 − 2 − 3 − = q 1 ;2 ; ;3 ℓ= ; ; ; = qc 1 ; ; 2 ∗; 3 ℓc =; ; ∗; (1.78) − − SU C SU L SU R U B L ⊗ ⊗ 1 ⊗ − 1 where = (3 2 1 + 3), = (1 2 1 1), = (3 1 2 3 ), = (1 1 2 +1), under (3) (2) (2) (1) . Given this embedding of the fermion fields one B L readily verifies that the electricQ chargeTL formulaTR takes− : the expression 3 3 = + + (1.79) 2 ∗ Next we have to state theδL Higgs; sector.; ; Inδ theR early; ; days; of the development; ; ; of left-right theories the breaking to the SM was minimally achieved by employing the following set of representations: = (1 2 1L +1), ; ; =; (1 1 2 +1)R and Φ; =; (1; 2 2 0) [2, 77, 78].δL However,δR as pointed out in [30, 34], in order to understand the smallness of neutrino i masses it is better to consider ∆ = (1 3 L;R1 +2) andL;Rσi ∆/ = (1 1 3SU+2)L;R in place of and . σ ∗σ ≡ ChoosingSU the matrixL SU representationR ∆ = ∆ 2 for the (2) adjoint and ˜ 2 2 defining the conjugate† doublet Φ Φ† , the transformation† properties for† the Higgs L UL L U ; R UR R U ; UL U ; UL U ; fields underÏ (2)L and Ï(2) read R Ï R Ï R ∆ ∆ ∆ ∆ Φ Φ Φ˜ Φ˜ (1.80) and consequentlyδL L TL; weL haveδL R δL TL δL TL 3 3 3 δR∆L = ∆ δR∆R =0TR; R δRΦ = ΦTR δRΦ˜ = Φ˜TR − − 3 3 3 δB−∆L =0L L δB−∆L =R ∆ R δB−ΦL = Φ δB−Φ˜L = Φ˜ : (1.81)

∆ = +2∆ ∆ = +2∆ Φ=0 Φ=0˜

Then, given the√ expression for the electric charge operator in Eq.∗ (1.79), we can decompose these ﬁelds in the charge eigenstates L;R / √ φ− φ φ − −φ∗ + ++ L;R ; 0 + ; 0 + : − / φ1 φ1 −φ2 φ 2 ∆ 0 2 ∆+ 0 0 ∆ = Φ= 2 2 Φ=˜ 1 1 (1.82) ∆ ∆ 2 Z

2 In order to ﬁx completely the theory one has to specify the action of the symmetry on the ﬁeld content. There are two phenomenologically viable left-right discrete 1.3. The Pati-Salam route ZP ZC 35

c 2 ψL2 ψR ψL ψL symmetries: and .ÎÏ They are defined as ÎÏ ∗ P L R C L R Z  ÎÏ † Z  ÎÏ T :    µ ÎÏ µ  µ ÎÏ µ∗ 2 W∆L W∆ R 2 W∆L W∆R : ÎÏ and : ÎÏ (1.83)  Φ Φ  Φ Φ   CP ZP ZC c ∗ PTheψ implicationsL ψR C of thisψL twoψL casesCγ differψR by the tiny amount of violation. Indeed Ï Ï ≡ 2 2 when restricted to the fermion fields we can identifyCP and respectively with P C 0 Z : Z and : . In the former case the Yukawa matrices are C hermitian whileZ in the latter they are symmetric. So if is conserved (real couplings) 2 2 C and lead to the same predictions. Z 2 Notice that involves an exchange between spinors with the same chirality. In 2 principle this would allow the embedding of into a gauge symmetry which com- C mutes with the Lorentz group. TheZ gauging is conceptually important since it protects theSO symmetry from unknown UV effects. 2 Remarkably itSO turns out that can be identified with a finite gauge transformation C of (10) which, historically,Z goes under the name of D-parity [96, 97, 98, 99, 100]. The connection with (10) motivates our notation in terms of left-handed fermion fields 2 which fits better for the case. Let us consider now the symmetry breakingv sector. From Eq.v (1∗.82) we deduce that the SM-preservingL;R vacuum directions are ; ∗ : h i vL;R h i v v 1 2 0 0 0 D E 0 ∆ = Φ = 2 Φ˜ = 1 (1.84) 0 0 0 vL vR The minimization of the scalar potential (see e.g. Appendix B of Ref. [34]) shows that be- vL vR ; vLvR γv ; v ; side the expected6 left-right symmetric minimum = , we have also the asymmetric one γ 2 1 2 ∗ P = = (in theL approximationR L R = 0) (1.85)Z C ↔ ↔ whereZ is a combination of parameters of the Higgs potential. Since the discrete 2 left-rightvR symmetryv vL is defined to transform ∆ ∆ (∆ ∆ ) in the case of 2 ≫ ≫ ( ), the VEVs in Eq. (1.85) breaks it spontaneously.v Phenomenologically we have to 1 R SU C SU L SU R U B L Z SU C SU L U Y require ⊗ ⊗ which⊗ leads to− the⊗ followingÊÏ breaking⊗ pattern⊗ v ≫vL 2 SU C U Q ; (3) (2) (2) (1) (3) ÊÏ1 (2) ⊗(1) MW ;MZ MW ;MZ (3)R R ≫(1) L (1.86)L MWR MZR where the gauge hierarchy is set by the gauge boson masses . i i B L Let us verifyD thisµ R by computing∂µ R igR TR; andR AR .µ Weig startB−L from− theR A covariantB−L µ ; derivative ∆ = ∆ + ∆ + ∆ ( ) (1.87) 2 SO

Chapter 1. From the standard model to 36 (10)

† µ Dµ R D R ; and the canonically normalized kinetich termi h i

Tr ( ∆ ) ∆ (1.88) M g v ; M g g v ; M ; which leads to WR R R ZR R B−L R Y 2 2 2 2 2 2 2 = = 2( + ) =0 (1.89)

± AR iAR gRAR gB−LAB−L gB−LAR gRAB−L whereWR ∓ ; ZR ; Y − : 1 √ 2 3 3 gR gB−L gR gB−L + = = q 2 2 = q 2 2 (1.90) − − − 2 gY gR gB−L + Z + gR gL g 2 2 2 11 ≡ 2 Given the relation = + gand the symmetry in Eq. (1.83) which implies = , we obtain MZR MWR : MWR : g 2gY ∼ 2 − 2 2 22 2 = 2 6 L (1.91) v h i 6 h i 6 At the next stage of symmetry breaking ( Φ = 0 and ∆ = 0) an analogous calculation M g v v ;2 M g g v v ; M ; yields (inWL the approximationL = 0)ZL Y L A 2 2 2 2 2 2 2 2 2 2 1 1 1 1 = +2 = + +4 =0 (1.92) 2 2 ± AL iAL gLAL gY AY gY AL gLAY where WL ∓ ; ZL − ; A : 1 √ 2 g3 L gY g3 L gY + = ρ = p 2 2 = p 2 2 (1.93) 2 + + M g g Notice that in order to preserve ρ= 1 atW treeL level,Y ; where 2 g 2 ≡ MZL 2 2 +2 vL v (1.94) ≪ MWL MWR SU L U Y 1 ⊗ one has to require . On the other hand at energy scales between and , (2) (1) is still preserved and Eq. (1.79) implies T B L : R − − 3 1 R SU R T∆R = ∆( ) B L (1.95) 2 − B L 3 Since ∆ is an (2) triplet ∆ = 1 andB−L we get a violation of by two units. Y TR Then11 two classes of and violating3 processes can arise: This relation comes directly from = + 2 (cf. Eq. (1.79)). For a formal proof see Sect. 2.2.4. 1.3. The Pati-Salam route B L 37 • B L • ∆ = 0 and ∆ = 2 which imply Majorana neutrinos. ∆ = 2 and ∆ = 0 which lead to neutron-antineutron oscillations. Let us describe the origin of neutrino masses while postponing the discussion of neutron-antineutronT oscillationsc T to thec next section.T cT ∗ c ν Y ℓ C ℓ Y ℓ C ℓ Y ℓ C Lℓ ℓ C R ℓ ; TheL piece⊃ of lagrangian relevant for neutrinos is Φ 2 Φ 2 SU ∆ L SU2 R 2 Φ + ˜ Φ˜ + ⊗ ∆ + ∆ + h.c. c (1.96)c ℓ UL ℓ ℓ UR ℓ T † Ï Ï The invarianceUL;R ofU Eq.L;R (1.96) under the (2) (2) might not be obvious. So let us recall that, on top of the transformation properties in Eq. (1.80), , , 2 2 and = T. Afterc projectingT c Eq. (1.96)T on the SMcT vacuumc ∗ directions and ν Y ν Cν v Y ν Cν v Y ν Cν vL ν Cν vR : taking onlyL the⊃ pieces relevant to neutrinos we get Φ 2v Φ 1 ∆ ˜ + ν νc+ + + h.c. (1.97) 2 Let us take for simplicity = 0 and consider real parameters. Then the neutrino Y vL Y v mass matrix in the symmetric basis ( T ) reads ; Y v Y vR ∆ Φ 1 ˜ vR v vΦL 1 ∆ (1.98) ≫ ≫˜ 1 and, given the hierarchy ρρ,T the matrixρ in Eq. (1.98) is block-diagonalized by a similarity transformation involving− T the orthogonalT ; matrix ρ1 ρ ρ −2 − − 1 1 ρ Y Y v /vR 2 ρ (1.99) 1 O 1 2 Φ ∆ 1 − T v where = ˜ . The diagonalizationmν Y vL Y isY validY up to: ( ) and yields − vR2 1 1 ∆ Φ ∆ Φ = ˜ ˜ (1.100) vL γv /vR The two contributions go under the name12 of type-II and type-I seesaw respectively.2 vL 1 From12 the minimization of the potential (see Eq. (1.85)) one gets = and Even without performing the complete minimization† we can† estimate† the induced VEV by looking V M L λ R : at the following piece of potential ⊃− L L L 2 ∆ Tr ∆ ∆ + Tr ∆ Φ∆˜ Φ (1.101) V M vL λvLvR v ; On the SM-vacuum Eq. (1.101) reads h i⊃− L | | 2 2 2 vL 1 ∆ + (1.102) vR v and from the extremizing condition with respectvL λ to | we| : get M 2 1L 2 = ∆ (1.103) SO

Chapter 1. From the standard model to 38 (10)

− T v hence the effective neutrinom massν matrixY γ readsY Y Y : − vR2 1 1 ∆ Φ ∆ Φ = ˜ ˜ (1.104) V A vR V A mν ThisÏ∞ equation is crucial since− it shows a deep connection between the smallness of neutrino masses and the non-observation of + currents [30, 34]. Indeed in the limit we recover thevR structure and vanish. Nowadays we know that neutrino are massive, but this information is not enough in order to fix the scale because theV detailedA Yukawa structures are unknown. In this respect one can adopt two complementary approaches. From a pure phenomenological P C point of view one can hope that the + interactions are just behind the13Z cornerZ and experiments such usM theWR LHC are probing rightKL nowKS the TeV region . Depending P − C 2 2 MonW theR & choice of the discreteZ left-rightMWR symmetry& : which can be eitherZ or , the strongest bounds on are given by the mass difference which yields 2 2 4 TeV in the case of and 2 5 TeV in the case of [101, 102]. Alternatively one can imagineSO some well motivated UV completion in which the Yukawa structure of the neutrino mass matrix is correlated to thatv ofR the charged fermions. For instance in (10) GUTs it usually not easy to disentangle the highest eigenvalue in Eq. (1.104) from14 the top mass. This implies that the scaleSO must be very heavy, somewhere close to 10 GeV. As we will see in Chapter 2 this is compatible with unification constraints and strengthen the connection between (10) and neutrino 1.3.2masses. Lepton number as a fourth color

One can go a little step further and imagine a partial unification scenario in which quarks and leptons belong to the same representations. The simplest implementation u u u ν dc dc dc ec is obtainedQ by collapsing the multipletsQc in Eq. (1.78) in the following way d d d e uc uc uc νc 1 2 3 − 1 − 2 − 3 − 1 2 3 SU =C U B−L SU C = 1 2 3 Q (1.105); ; c ⊗ ∗ ⊂ Q ; ; SU C SU L SU R ⊗ ⊗ so that (3) (1) (4) and the fermion multiplets transform as = (4 2 1) SU L SU R and ↔= (4 1 2 ) under (4) (2) (2) , which is known as the Pati-Salam group [2]. Even in this case one can attach an extra discrete symmetry which exchange (2) (2) . L ; ; R ; ; The Higgs sector of the model is essentially an extension of that of the left-right symmetric model presented in Sect. 1.3.1. IndeedO we have ∆ = (10 3 1), ∆ = (10 1 3) 13 It has been pointed out recently [36] that a low (TeV) left-right symmetry scale could be welcome in view of a possible tension between neutrinoless double beta decay signals and the upper limit on the sum of neutrino masses coming from cosmology. 1.3. The Pati-Salam route ; ; ∗ / / 39 ⊕ − ⊕ − SU C SU C U B L ⊃ ⊗ − R and Φ = (1 2 2 ). From the decompositionh i 10 = 6(+2 3) 3( 2 3) 1( 2) under (4) (3) (1) and the expression for the electric charge operator in Eq. (1.79), we can readily see that ∆ contains a SM-single direction and so the first h Ri stage of the breakingSU C isSU givenL bySU R SU C SU L U Y ; ⊗ ⊗ ÊÏ∆ ⊗ ⊗ SU C U Q (4) (2) (2)⊗ (3) (2) (1) (1.106) h i L whileh i≪h the finali breaking to (3) (1) is obtained by means of the bi-doublet VEV Φ . Analogously to the left-right symmetric case an electroweak triplet VEV ∆ Φ is induced by the Higgs potential and the conclusions about neutrino masses are the same. SU C SU C / / A peculiar feature of the Pati-Salam model is that the proton⊕ is stable⊕ − in spite⊕ of SU C U B L the quark-lepton⊗ transitions− due to the (4) interactions. Let us consider first gauge XPS XPS interactions.≡ The adjoint≡ of − (4) decomposes as 15 = 1(0) 3(+4 3) 3( 4 3) 8(0) 4 4 under (3) (1) . In particular the transitionsPS between quark and leptons due 3 g PS µ 3 µ c µ c c µ c to 3(+PS ) and Xµ uγ3( ν ) comedγ e fromXµ theu currentγ ν d interactionsγ e L ⊃ √ + + + G +h.c.G XPS (1.107) 2 c c c− G u G d G ν G e G XPS G u G d G ν 2 c − 3 GIt turnse out thatG Eq.1 (1.107) hasB an accidentalL global symmet2 ry , where ( 1 ) = , −B L 3 3 B L 3 ( ) = ( −) = + , ( ) = ( ) = +1, ( ) = + , ( ) = ( ) = , ( ) = ( ) = 1. is nothing but + when evaluated on the standard fermions. Thus, given that is also a (gauge) symmetry, we conclude that both and are conserved by the gauge interactions. B The situation regardingqqqℓ the scalar interactions is more subtle. Actually in the min- qqqℓ QQQQ Q imal model there is an hidden⊂ discrete symmetry which14 forbids all the ∆ = 1 transitions,ijkl like for instance (seeSU e.g.C Ref. [103] )4 . A simple way to see it is that any operator of the type and the term must be contractedSU C with an tensor in orderQ to form an (4) singlet. However, since the Higgs fields in the minimal model are4 eitherB singlets or completely symmetric in the (4) space, they cannot mediate operators. d On the other hand ∆ = 2 transitions like neutron-antineutron oscillations are R allowed and they proceed throughh =i 9udd operatorsudd ; of the type [103] M R ∆6 ∆ ( )( ) R R(1.108) h i h i which are generated by the Pati-Salam breaking VEV ∆ .SU The fact that ∆ B can be14 pushed down relatively close to the TeV scale without making the proton to decay Notice that this is just the reverse of the situation with the minimal (5) model where ∆ = 2 transitions are forbidden. SO

Chapter 1. From the standard model to 40 (10) is phenomenologicallyτN > interesting, since one can hope in testable neutron-antineutronτn−n > oscillations (for32 a recent review see Ref. [104]). Present bounds on nuclear instability ¯ give8 10 yr, which translatesB into a bound on the neutron oscillationd time 10 sec. Analogous limits come from direct reactor oscillations experiments. This sets a lower bound on the scale of ∆ = 2 nonsupersymmetric ( = 9) operators that varies from 10 to 300 TeV depending on model couplings. Thus neutron-antineutron 1.3.3oscillations One probe family scales uniﬁed far below the uniﬁcation scale.

The embedding of the left-right symmetric models of the previous sections into a grand unified structureSU requiresSO the presence of a rank-5 group. Actually there are only two candidates which have complex representations and can contain the SM asF a subgroup.F SU U ⊕ These are15 ⊗(6) and (10). The former group even though smaller it is somehow redundant since the SM fermions would be minimally embedded into 6 15 which ; under (5) (1) decompose⊕ − as − ⊕ 6 = 1(+5) 5( 1) and 15=5( 4) 10(+2) (1.109) SO yielding an exotic 5 on top of the SM fermions. SU U SO SO SU SU SU Thus⊗ we are left with⊗ (10). There are essentially two ways of⊗ looking⊗ at this unified theory, accordingSO to the two maximal subalgebras which contain the SM: (5) (1) and (6) (4). The latter is locally isomorphicSO to (4) (2) (2). SU U ; ; ; ; The group theory of − (10)⊕ will be⊕ the− subject of the next⊗ section, but let us alread⊕ y an- SU C SU L SU R ticipate that the⊗ spinorial⊗ 16-dimensional representation of (10) decomposes in the following way 16 = 1( 5) 5(+3) 10( 1) under (5) (1) and 16 = (4 2 1) (4 1 2) under (4) (2) (2) , thus providing a synthesis of both the ideas of Georgi- Glashow and the Pati-Salam. 1.4 O group theory S SO O (10) T (10) is the special orthogonal groupφ of rotationsO inφ a 10-dimOφ ensional vectorφ φ space. Its T Ï Odefining representationOO is given by the group of matrices Owhich leave invariant the normSU of a 10-dimentional real vector . Under , and since is invariant must be orthogonal, = 1. Here special means det = +1 which selects the 15 (6) as a grand unified group deserves anyway attention especially in its supersymmetric version. SU SU The reason is that it has an in-built mechanism in which the doublet-triplet splitting⊗ can be achieved in a very natural way [105, 106]. The mechanism is based on the fact that the light Higgs doublets arise as pseudo-Goldstone modes of a spontaneously broken accidental global (6) (6) symmetry of the Higgs superpotential. SO

1.4. group theory (10) 41O Tij Tji i;j ;::: − group of transformations continuously connected with the identity. The matrices O T ; may be written in terms of 45 imaginary generatorsij ij = , for =1 10, as 1 2 ij = exp (1.110)

Tij ab i δa iδbj ; where are the parameters of the transformation.− A convenient basis for the generators is a;b;i;j ;::; [ ] SO ( ) = ( ) (1.111) where = 1 10 and the square bracket16 stands for anti-symmetrization. They Tij ;Tkl i δikTjl δjlTik δilTjk δjkTil : satisfy the (10) commutation relations − − SO = ( + ) (1.112) OOT O In oder to study the group theoryδij of (10) it is crucial to identify the invariant tensors. T TheOO conditions = 1 and det = +1 give rise to two of them. The first one is δij OikOjlδkl OikOjk δij ; simply the Kronecker tensor Ï which is easily proven to be invariant because of = 1, namely ijklmnopqr = = (1.113) while the second one is the 10-index Levi-Civita tensor . Indeed, from the O O O O O O O O O O O definition ofi′j determinant′k′l′m′n′o′p′q′r′ i′i j′j k′k l′l m′m n′n o′o p′p q′q r′r ijklmnopqr O det = ijklmnopqr (1.114) SO and the fact that det = +1, we conclude that is also invariant. The irreducible representations of (10) can be classified into two categories, single-valued and double-valued representations. The single valued representations have the same transformations properties as the ordinary vectors in the real 10- dimensional space and their symmetrized or antisymmetrized tensor products. The doube-valued representations, called also spinor representations, trasform like spinors 1.4.1in a 10-dimentional Tensor representations coordinate space. n SO

The generaln -index irreducible representations of (10) are built by means of theφi antisymmetrization or symmetrization (including trace subtraction) of the tensor prod- φi Oij φj ; uct of -fundamental vectors. Starting fromÏ the 10-dimentional fundamental vector , whose transformation rule is SO J ;J iJ J T J T J T (1.115) 16 ≡ ≡ ≡ These are an higher dimensionalTij generalizationTkl of the well known (3) commutation relations 1 2 3 1 23 2 31 3 12 [ ]= , where , and . Then the right hand side of Eq. (1.112) takes just into account the antisymmetric nature of and . SO

Chapter 1. From the standard model to 42 (10)

δij δij we can decomposeφi φ thej φ tensorj φi productφi ofφj twoφj of themφi inφ thek folloφk wingφk wayφk φi φj ⊗ − ⊗ ⊗ ⊗ − ⊗ ⊗ : ⊗ 1 φA 1 φS Sδ ( ij )+ ( + ij ) + ij = |2 {z } |2 {z 10 } |10 {z } (1.116) A φij S Since theφij symmetry properties of tensors under permutation of the indices are not changed by the group transformations, the antisymmetricO tensor and the symmetric A S tensor clearly do not transform into each other. In general one canφ aijlsoφij separateSδij a tensor in a traceless part and a trace. Because is orthogonal also the traceless/ / − property is preserved− by the group transformations. So we conclude that , and form irreducible representations whose dimensions are respectively 10(10 1) 2 = 45, 10(10 + 1) 2 1 = 54 and 1. One can continue in this way by considering higher order representations and separating each time the symmetric/antisymmetric pieces and subtracting traces. ijklmnopqr However something special happens for 5-index tensorsφ andnopqr the reason has to do with the existence of the invariant i which induces the following duality map when applied to a 5-indexφijklm completelyφijklm antisymmetricijklmnopqr tensorφnopqr : Ï ≡ − φ ˜ ijklm (1.117) 5! This allows us to define the self-dual and the antiself-dual components of in the ijklm φijklm φijklm ; following way ≡ √ 1 Σijklm φijklm + φ˜ijklm : (1.118) ≡ √2 − 1 ˜ ijklm Σijklm ijklm ijklm (1.119) 2 − ijklm ijklm ˜ One verifies that Σ˜ = Σ (self-dual) and Σ = −Σ (antiself-dual). Since the duality property is not changed by the group transformat1 10! ions Σ and Σ do 2 5!(10 5)! 1.4.2form irreducible Spinor representations whose dimension is = 126. SO x ; x ;:::;x x x ::: x We have defined the (10) group by those linear2 transformations2 xi 2 on the coordinates 1 2 10 1 2 10 , suchx thatx the::: quadraticx formγ x γ+ x +::: +γ xis left; invariant. If we write this quadratic form as the square of a linear form of ’s, 2 2 2 2 1 2 10 1 1 2 2 10 10 + + + = ( + + + ) (1.120) γi; γj δij : { } we have to require =2 (1.121) SO

1.4. group theory (10) γ 43

Eq. (1.121) goes under the name of Clifford17 algebra and the ’s haveγ to be matrices in orderSO N to anticommute with each other .N For definiteness let18 us build an explicit representation of the ’s which is valid for (2 ) groups [107] . We start with = 1. Since the Pauli matrices satisfy the σi;σj δij ; Clifford algebra { } =2 (1.125)

i we can chooseγ σ γ σ : i − (1) (1) 1 1 0 1 2 2 0 = = and = = (1.126) N > 1 0 0 N N

Then the case 1 is constructedN by recursion. The iteration from to +1 is N γi deﬁned by γi N i ; ;:::; N; ( ) γi ! ( +1) − 0( ) = for =1 2 2 (1.127) 0 i γ N γ N : N N i − ( +1) ( +1) 2 +1 0 1 2 +2 0 =N and = (1.128) γi 1 0 0 ( ) Given the fact that the matrices satisfy the Cliﬀordγ algebra let us check explicitly 17 In particular it can be shown that the dimension of the matrices must be even. Indeed from Eq. (1.121) we obtain γj γiγj γj γi γj γj γiγj γi ;

j ( + ) = 2 or = (1.122) with no sum over . Taking the trace we get γj γiγj γi :

i j Tr = Tr (1.123) 6

But for the case = this implies γj γiγj γiγj γj γi ; − −

Tr = Tr = γiTr γi (1.124) γi γi − 2 γi and− hence, putting together Eqs. (1.123)–(1.124), we have Tr = 0. On the other hand, = 1 implies that the eigenvalues of are either +1 or 1. This means that to get Tr = 0, the number of +1 and 118 eigenvalues must be the same, i.e. must be even dimensional. For an alternative approach to the construction of spinor representations by means of creation and annihilation operators see e.g. Ref. [108]. SO

Chapter 1. From the standard model to 44 N (10) γi ( +1) γ N ; γ N that the N onesN satisfy iti as well,j δij γi ; γj N N δij ;  n ( ) ( )o γ ; γ  δij n ( +1) ( +1)o j i ( )0 ( ) 2 0 =  n No  = N =2 (1.129) N N γi 0 2 γi γi ; γ N 0 N N − ; γi ( ) ! γi ( ) ! n ( +1) ( +1)o − 2 +1 0( ) 0( ) = + N =0 (1.130) γ N : 0 0 2 ( +1) 2 +1 =1 (1.131)

N N N N N Analogouslyγi one; γ N ﬁnds δij ; γ N ; γ N ; γ N : 2 n ( +1) ( +1)o n ( +1) ( +1)o ( +1) 2 +2 2 +1 2 +2 ′ 2 +2 =2 x=0i Oikxk O=1 (1.132) γi Now consider a rotation in the coordinate space, = , where is an orthogonal γ′ O γ : matrix. This rotation induces a transformationi ik k on the matrix

= (1.133) ′ ′ γ ; γ OikOjl γk; γl δij : Notice that the anticommutation{ i relationsj } remain{ } unchanged, i.e. γ = =2 (1.134) γ Because the original set of matrices form a complete matrix algebra, the new set of γ′ S O γ S− O O γ S O γ S− O : matrices must bei related toi the original set byik ak similarityi transformation, 1 1 O S O N = (Ï) ( ) or = ( ) ( ) (1.135) ψi The correspondence ( ) serves as a 2 -dimensional representation of the rota- ′ tion group which is called spinor representation.ψi S O ij ψj ; The quantities , which transform like Oik S O = ( ) (1.136) O δ S O iS ; are called spinors. For anik inﬁnitesimalik ik rotation we can paraijmetrizeij and ( ) by 1 ik ki 2 − = + and ( )=1+ (1.137)

i Skl; γi γlδik γkδil ; with = . Then Eq. (1.135) implies −

ikγk lkγkδil γkδil γkδjl [ ] = ( − ) (1.138) Skl 1 2 i where we have used = Skl= ( γk; γl : ). One can verify that a solution for in Eq. (1.138) is = [ ] (1.139) 4 SO

1.4. group theory (10) 45 kl S O π S O By19 expressing theSO parameterN in terms of rotationsS angle,O one can see that ( (4 )) = 1 , i.e. ( ) is a double-valued representation.γχ

However for (2 ) groups the representationN ( ) is not irreducible. To see this γχ i γ γ γ N : we construct the chiral projector deﬁned− by··· 1 2 2 γχ γi N= ( ) γχ; Skl(1.141) ψ ψ′ S O ψ i 20 ij j anticommutes with since 2 is even and consequently we get = 0 − (cf. Eq. (1.139)). Thusψ if transformsγχ ψ as = ( ψ) , the positiveγχ ψ and negative chiral ≡ ≡ − components + 1 ψ ψ− 1 N−(1 + ) and (1 ) (1.142) 2 + 2 − transform separately. In other1 wordsψ andψ form two irreducible spinor representations of dimension 2 . + ψ T SOWhichN is the relation betweenψ Cψ SOand N ? In order to address this issue it is necessary to introduce the conceptT of conjugation. Let us consider a spinor of S C CSij : (2 ). The combination is anij (2− ) invariant provided that C = C iσ(1.143) N (1) N 2 N C The conjugation matrix Ccan be constructedN iteratively.N We: start from = for C ( ) = 1 and deﬁne ( +1) − ( +1)0 ( ) = (1.144) N − ( T) N N 0 C γ C γi : i − One can verify that ( ) 1 ( ) T γi ( ) = ( ) (1.145) N T − N By transposing Eq. (1.145) and substitutingγi; C backC we: get h ( ) 1 ( )i (( ) ) =0 (1.146)

N T − N N N T Then the Shur’s LemmaC impliesC λ I C λ C ; ( ) 1 ( ) ( ) ( ) SO (( ) ) = or = ( ) (1.147) 19 This is easily seen for (3). In this case the Clifford algebra is simply given by the three Pauli i ' σi'i ' matrices and a finite transformationS O ' lookse likeσi'i | | i | | ; ' 2 | | ( ( )) = = cos + sin (1.140) ' ' '2 ' '2 ' ' ≡− ≡− ≡− | | q SO N 2 2 2 23 1 13 2 12 3 1 2 3 where20 we have defined , , and = + + . Notice that this would not be the case for (2 + 1) groups. SO

Chapter 1. From the standard model to 46 λ λ λ (10) − 2 CT N N / C: which yields = 1. In order to choose− between = +1 and = 1 one has to apply Eq. (1.144), obtaining ( +1) 2 = ( ) (1.148) N − T N N C γ C γχ ; On the other hand Eq. (1.141) and Eq. (1.145)χ lead− to ( ) 1 ( ) γT γ χ χ ( ) = ( ) (1.149)

N − N N C γχC γχ : which by exploiting = (cf. again Eq. (1.141))− yields ( ) 1 ( ) ( ) = ( ) (1.150) N − ∗ N N − ∗ N N C Sij γχ C C S γχ C Sij γχ : This allows us to write ij − − ( ) 1 ( ) ( ) 1 ( ) γ ( ) ( (1 + )) = ( ) (1 + ) = 1 + ( ) (1.151) SO N N ψ ψ− where we have also exploited the hermicityC of the matrices.+ Eq. (1.151) can be in- − terpretedSO in theN followingN way:ψ for (2 ) with evenψ and SOarek self-conjugate i.e. real or pseudo-real depending+ on whetherSO is symmetric or antisymmetric (cf. Eq. (1.148)), while for (2 ) with odd is the conjugate of . Thus only (4 +2) can have Spinorscomplex will representations be spinors and remarkably (10) belong to this class.

SO N SO N′ SO N ⊂ We close this section by pointingγ outχ a distinctive feature of spinorial representations: spinors of (2 ) decompose into the directN sum of spinors of (2 ) (2 ) [107]. Indeed, since the construction ofN in Eq.γχ (1.141) is such that γχ N ; ( ) γχ ! ( +1) − 0( ) = M− (1.152) ψ SO N 0 M M− + SO N 1 the positive-chirality1 spinor of (2 +2 ) contains 2 positive-chirality spinors ψ ψ ψ− andSO 2 N Mnegative-chiralityÏ SO N M− spinors⊕ SO ofN M−(2 ). More explicitly + + ψ ψ− (2 +2 ) (2 Ï+2 ×2) SO N (2M−+2 ⊕2) × SO N M− Ï··· + M− ψ M− ψ− : (2 +2 4) Ï ×(2 SO+2 N 4) ⊕ × SO N 2 2 1 + 1 (2 ) (2 ) 2 2 (1.153) SO N ;M N ;M Let us exemplify this important concept in the case of the 16-dimensional positive- chirality spinor of (10).− By takingSO respectivelySO ( = 3 = 2) and ( = 2 = 3) we• obtain × ⊕ × ⊃ + 16=2 4 2 4 under (10) (6), SO

1.4. group theory (10) − SO SO 47 • × ⊕ × ⊃ − + − 16=4SO 2 SO4 2 under (10) (4), SO SO SO + + ⊃ ⊗ where 4 (4 ) and 2 (2 ) are respectively the positive (negative) chiral components ; −; − : of the (6) and (4) reducible spinors. Thus⊕ under (10) (6) (4) the 16 decomposes as + + SO SO SU SU SU 16 = (4 2 ) (4 2 ) (1.154) ⊗ As weSO will showSU in Sect. 1.4.4 the Lie algebras (6) and (4) are locally isomorphic to (4) and (2) (2). This allows us to make− the; following identifications between the (6) and (4) representations∼ ∼ SO SU SU + ⊗ 4 4 4 4 (1.155) ; − ; ; and the (4) and (2) (2)∼ ones ∼ + SO 2 (2 1) 2 (1 2) (1.156) SU C SU L SU R ⊗ ⊗ which justify the decomposition of the; ; (10); spinor; : under the Pati-Salam algebra (4) (2) (2) as anticipated in Sect.⊕ 1.3.3, namely 16 = (4 2 1) (4 1 2) (1.157) This striking group-theoretic feature of spinors, which under the natural restriction to an orthogonal subgroup decompose into several copies of identical spinors of the subgroup,SO hints to a suggestive connection with the repetitive structure of the SM families [107] and motivatesSO theSO study of unification in higher orthogonal groups than (10) [27, 107, 109, 110]. To accommodate at least the three observed matter families we must use either (16) or (18). Following− the decomposition in Eq. (1.153) we SO ψSO ψSO ψSO get• Ï × ⊕ × + + − SO ψ (16) ψ (10) ψ (10) • (16): SO Ï 4 × SO ⊕ 4 × SO , + + (18) (10) (10) (18): 8 8 SO k. However there is a fundamental difference between the two cases above. AccordingψSO to the discussion below Eq. (1.151) only (4 + 2) groups have complex spinor+ rep- (16) resentations. This meansψSO that one can write a super-heavy bare mass term for and it is difficult to explain+ why it should be light. On the other hand no bare mass (18) term can be written for , making the last group a more natural choice. The obviousψSO difficulty one encounters inψ thisSO class of models is the overabundance − ψofSO sequential+ or mirror families. If we decide+ to embed the SM fermions into three (10) (10) copies of , the21 remaining families in are called sequential, while those in (10) are mirror families. 21 Mirror fermions have the identical quantum numbers of ordinary fermions under the SM gauge group, except that they have opposite handedness. They imply parity restoration at high-energies as proposed long ago by Lee and Yang [111]. SO

Chapter 1. From the standard model to 48 (10)

It has been pointed out recently [112] that the existence of three (mirror or sequential) familiesSO is still inSO accord with the SM, as long as an additional Higgs doublet is also present. This however is not enough to allow large orthogonal uniﬁcation scenarios based1.4.3 on Anomaly(16) or cancellation(18). SO SO Tij (10) is an anomaly-free group.T Thisij important property can be understood from a simple groupi j theoretical argument [113]. Let us consider the (10) generators in a given arbitrary representation. transforms like an antisymmetric tensor in the Tij ;Tkl Tmn ; indices and . Then the anomaly, which{ is proportional} to the invariant tensor δ Tri j k l m n (1.158) ↔ ↔ ↔ must be a linear combinationij kl kl of amn productij of Kroneckermn ’s. Furthermore it must be ↔ ↔ i j k ↔ l m n antisymmetric under the exchanges ↔ , ↔ , ↔ and symmetric under the exchange of pairs , and . However the most general form δjkδlmδni δikδlmδnj δjlδkmδni δilδkmδnj δjkδlnδmi δikδlnδmj δjlδknδmi δilδknδmj ; consistent− with the antisymmetry− in ,− , − ij kl SO ↔ + + + ijklmn is antisymmetric in SOas well and so it mustSU vanish. The proof fails for (6) where the anomaly can beSO proportionalN to theN >invariant tensor . Actually this is consistent with the fact that (6) is isomorphic to (4) which is clearly an anomalous 1.4.4group. On The the standard other hand model( ) is embedding safe for 6. SO

From the (10) commutationT relations; T ; T in; Eq. T (1.112); T : we ﬁnd that a complete set of simultaneously commuting generators can be chosen as 12 34 56 78 90 (1.159)

This is also known as the CartanTC; subalgebra TC; TL; TR and; TB can−L : be spanned over the left-right group Cartan generatorsSO SO 3 8 3 3 SO ⊗ SO Tij i;j ; ; ; SO T(1.160)ij i;jLet us; consider; ; ; ; the (4) SO (6) maximal subalgebra of (10). We can span the SO SO (4) generators over with = 1 2 3 4 and the (6) generators⊗ over with =5 6 7 8 9 0. From the (10) commutation relations in Eq. (1.112) one can verify SO SU SU SO SU SO that these∼ two sets⊗ commute (hence∼ the direct product (4) (6)). The next information we need is the notion of local isomorphism for the algebras TL;R T T ; TL;R T T ; TL;R T T ; (4) ≡ (2) ± (2) and (6) ≡ (4).± In the (4) case≡ we deﬁne± 1 1 2 1 3 1 2 23 14 2 31 24 2 12 34 ( ) ( ) ( ) (1.161) SO

1.4. group theory (10) 49

i j ijk k i j ijk k i j and checkT byL;T anL expliciti T calculationL ; TR that;TR i TR ; TL;TR : h i h i h i i i TL TR i= ; ; = SU L =0SU R (1.162) SO Thus and ( = 1 2 3) span respectively the (2) and the (2) algebra. On T T T ; T T T ; T T T ; the otherC ≡ hand for the (6) sectorC ≡ we deﬁne C ≡ 1 1 2 1 3 1 T 2 T89 T70 ; T 2 T97 T80 ; T 2 T09 T87 ; C ≡ ( + ) C ≡ ( + ) C ≡ ( + ) 4 1 5 1 6 1 T 2 T96 T05 ; T 2T 59 T 06 T ; T 2 T67 T85 ; C ≡ ( + ) C ≡ √ ( + ) C ≡ ( + ) 7 1 8 1 9 1 2 75 86 2 3 65 78 09 2 67 58 TC ( T + T ) ; TC (2 T + T + ;) TC (T + T ) ; ≡ ≡ ≡ 10 1 11 1 12 1 T 2 T75 T68 ; T 2 T69 T05 ; T T2 95 T 06 T ; C ≡ ( + ) C ≡ ( + ) C ≡ √ ( + ) 13 1 14 1 15 1 2 89 07 2 97 08 65 87 90 ( + ) ( + ) 6 ( + + )

i j ijk k and verify after a tedious calculationTC;T thatC i f TC ; h i ijk i f SU= TC i ;:::;(1.163) i SU C SU C TC i ;:::; TC TB L where are the structure constants of (4) (see e.g. [114]). Thus− ( =1 15) spans the (4) algebra15 and, in particular, the (3) subalgebra is spanned by ( = 1 8) while can be identiﬁed with the (normalized) generator. Then the hypercharge and electric charge operators read respectively Y TR TB L T T T T T − − 3 r 2 1 12 34 1 65 87 90 = + = ( ) + ( + + ) (1.164) 3 2 3 Q TL Y T T T T : and 3 12 1 65 87 90 = + = + ( + + ) (1.165) 1.4.5 The Higgs sector 3 SO

As we have seen in the previous sections (10) offers a powerful organizing principle for the SM matter content whose quantum numbers nicely fit in a 16-dimensional spinorial representation. However there is an obvious prize to pay: the more one unifies the more one has to work in order to break the enhanced symmetry. The symmetry breaking sector can be regarded as the most arbitrary and challenging aspect of GUT models. The standard approach is based on the spontaneous SO

Chapter 1. From the standard model to 50 (10) symmetry breaking through22 elementary scalars. Though other ways to face the problem may be conceived the Higgs mechanism remains the most solid one in termsSO of SU C U Q computability⊗ and predictivity. The breaking chart in Fig. 1.2 shows the possible symmetry stagesSO between (10) and (3) (1) with the corresponding scalar representations responsible for the breaking. That gives an idea of the complexity of the Higgs sector in (10) GUTs.

SO SU U X ⊗

(10) breakingSU chart with representationsSU up toC theSU 210. L (5)U Y (1) can be understood Figure 1.2: ⊗ ⊗ either in the standard or in the ﬂipped realization (cf. the discussion in Sect. 3.1.2 ). In the former case 16 or 126 breaks it into (5), while in the latter into (3) (2) (1) . For simplicity we are neglecting the distinctions due to the discrete left-right symmetry (cf. Sect. 2.1 for the discussion on the D-parity and Table 2.1 for an exhaustive account of the intermediate stages).

In view of such a degree of complexity, better weSO start by considering a minimal Higgs sector. Let us stress that the quest for the simplest Higgs sector is driven not 22 For an early attempt of dynamical symmetry breaking in (10) see e.g. [115]. SO

1.4. group theory (10) 51 only by aesthetic criteria but it is also a phenomenologically relevant issue related to tractability and predictivity of the models. Indeed, the details of the symmetry breaking pattern, sometimes overlooked in the phenomenological analysis, give further constraints on the low-energy observables such as the proton decay and the effective SM flavor structure. For instance in order to assess quantitatively the constraints imposed by gauge coupling unification on the mass of the lepto-quarks23 resposible for proton decay it is crucial to have the scalar spectrum under control . From the breaking chart in Fig. 1.2 we concludeSO that, before before considering any symmetry breaking dynamics, the following representations are required by the group theoryH in orderH to achieve a full breaking of (10) down toSU the SM: • 16 or 126 : they reduce the rank by one unit but leave an (5) little group

unbroken.H H H SU U • SU ⊗ 45 or 54 or 210 : they admit for little groups different from (5) (1), yielding the SM when intersected withSO (5). Ï H It should be also mentioned that a one-stepF (10)F SM breaking can be achieved via only one 144 irreducible Higgs representation [54]. However, such a setting requires an extended matter sector, including 45 and 120 multiplets, in order to accommodate realistic fermion masses [55]. As we will see in the next Chapters the dynamics of the spontaneous symmetry breaking imposes further constraints on the viability of the options showed in Fig. 1.2. On top of that one has to take into account also other phenomenologicalH constraintsH due to the unification pattern, the proton decay and the SM fermion spectrum. We can already anticipate atH thisH level that whileH the choice between 16 or 126 His a model dependent issue related to the details of the Yukawa sector (see e.g. Sect. 1.5), SU U H H H the simplest option among⊗ 45 , 54 and 210 is certainly given by the adjoint 45 . However, since the early 80’s, it has been observed that the vacuum dynamics aligns the adjoint along an (5) (1) direction, making the choice of 16 (or 126 ) and 45 alone not phenomenologically viable. In theSU nonsupersymmetric case the alignment is only approximate [56, 57, 58, 59], but it is such to clash with unification constraints (cf. Chapter 2) which do not allow for any (5)-like intermediate stage, while in the supersymmetric limit the alignment is exact due to F-flatness [60, 61, 62], thus never landingH to a supersymmetric SM vacuum. The critical reexamination of these two longstanding no-go for the setting with the 45 driving the GUT breaking will be the subject of Chapters 3 and 4. 23 Even in that case some degree of arbitrariness can still persist due to the fact that the spectrum can never be fixed completely but lives on a manifold defined by the vacuum conditions. SO

Chapter 1. From the standard model to 1.552 Yukawa sector in renormalizable SO (10)

SO (10) S A S ; In order to study the (10) Yukawa⊗ sector,⊕ we decompose⊕ the spinor bilinear S A 16 16 = 10 120 126 (1.166) where Hand H denote theH symmetric (S) and antisymmetricSO (A) nature of the bilinear couplings in the family space. At the renormalizable level we have only three possibil- Y F Y H Y H Y H F ; ities: 10 , 120 andL 126 . Thus the most general (10) Yukawa lagrangian is given by Y Y 10 120 126 Y = 16 10 + 120 + 126 16 + h.c. (1.167) 10 126 120 where 24 and are complex symmetricH H matrices while is complex antisym-SO metric . H H It should be25 mentioned that 10 and 120 are real representation from the (10) point24 of view . In spite of that the components of 10 and 120 can be chosen either For completeness we report a concise proof of these statements based of the formalism used in Sect. 1.4.2 and borrowed from Ref. [107]. In a schematic notation we can write a Yukawa invari- ψT C C ψ ; ant term such as those in Eq. (1.167) as D k k ψ SO 5 C C ( Γ )Φ D (1.168) SO k k γ 5 wherek is both a Lorentz and an (10) spinor (hence the need for kand which areSO respectively T ψ γχ γχ ; γi C γχ γχC γ γχ the Dirac and the (10) conjugation matrix).{ } Then Γ denotes− an antisymmetric productχ of matrices and Φk is a scalar field transforming like an antisymmetric tensor with indices under (10).k Using 5 5 thek facts; ; that is an eigenstate of , = 0, = − (cf. Eq. (1.150)) and− = , we deduce that must be odd (otherwise Eq. (1.168) is zero). This singles10! out the antisymmetric10! tensors Φ with = 1 3 5, corresponding respectively to dimensions 10, 3!(10 3)! = 120 and 5!(10 5)! = 252 (actually the duality map defined in Eq. (1.117) is such that only half of these 252 components couples to the spinor T T C CD C C bilinear). D − − Next we consider the constraintsT imposedT T byT theT symmetryT proper− tiesT of the conjugation matrices, ψ 5CDC 5kψ ψ C C ψ ψ CDC C C ψ ; namely = and = (cf. Eq.− (1.148)).D k These− yields k 1 5 5 5 5 5 − T Γ = Γ = ( Γ ) (1.169) C γ C γi i − where in the second− step we1 T have used− T the anti-commutationT k propertiesk ofk thek− / fermion fields. Then, by C γ γ5 k C 5 C γ γ C γk γ γ γk ; exploiting the relation· · · = k(cf.· · ·Eq. (1.145)),− we obtain· · · − − · · · 1 1 ( 1) 2 1 5 5 1 1 5 ( ) = 5 1 = ( ) = ( ) ( ) (1.170) T k k− / k T ψ CDC kψ ψ CDC kψ : which plugged into Eq. (1.169) implies − ( 1) 2+ +1 k ; 5 5 ψ k Γ = ( ) Γ (1.171)

Hence for = 1 3 the invariant in Eq. (1.168) is symmetricSO in the ﬂavor space of , while for = 2 is ∗ Ta T antisymmetric.25 − a This can be easily seen from the fact that theS (10) generators in the fundamental representation are both imaginary and antisymmetric (cf. Eq. (1.111)). This implies = which corresponds to the deﬁnition of real representation in Eq. (1.3) with = 1. SO

1.5. Yukawa sector in renormalizable (10) 53 H ∗ H ∗ 6 H 6 H real or complex. In the latter case we have 10 = 10 and 120 = 120 , which means that the complex conjugate ﬁeldsSO diﬀer from the original ones by some extra charge. Actually both the components are allowed26 in the Yukawa lagrangian, since Y F Y H Y ∗ Y H Y ∗ Y H F : theyL transform in the same wayH under (10) , andH thus we have 10 10 120 120 126 = 16 10 + ˜ 10 + 120 + ˜ 120 + 126 16 + h.c. (1.172)

For instance complex scalars are a must in supersymmetryY Y where the fundamental objects are chiral superﬁelds made of Weyl fermions and complex scalars. However 10 120 in supersymmetry we never see the couplings ˜ and ˜ because of the holomorphicH properties of theH superpotential. Even without supersymmetry there could be the phenomenological need, as we are going to see soon, of having either a complex 10 or a complex 120 . In this case the new structures in Eq. (1.172) areH still there, unless SU C SU L SU R an extra symmetry which forbids⊗ them⊗ is imposed. In order to understand the implications; ; of having; ; a: complex 10 , let us decompose it under the subgroup (4) (2) ⊕(2) SU C SU L U Y 10 = (1 2 2) (6 1 1) ⊗ ⊗(1.173) ; ; ; ; Hu ; ; Hd H ∗ H∗ Hd ≡ ⊕ − ≡ H u H ∗ H∗ Hd In particular the bi-doublet1 can6 beH furtheru1 decomposed6 under (3) (2) (1) , yielding (1 2 2) = (1 2 + 2 ) (1 2 2 ) . Now if 10 = 10 we have = as in the SM, while if 10 = 10 then = as much as in the MSSM or in the two-higgs doublet model (2HDM). Y ToY simplify a bit the discussion let us assume that we are eithHu∗er inHd the supersymmet- 6 10 ric case or in the nonsupersymmetric one with an extra symmetry which forbids ˜ ˜120 and , so that; Eq.; (1.167); ; applies; with complex bi-doublets ( = ). The remaining representations in Eq.⊕ (1.167) decompose as ; ; ; ; ; ; ; ; ; ; ; ; ; ⊕ ⊕ ⊕ ⊕ ⊕ 16 = (4 2; 1); (4 1;2); ; ; ; ; ; (1.174) 120 = (1 2 2) ⊕ (10 1 1) ⊕ (10 1 1) ⊕ (6 3 1) (6 1 3) (15 2 2) (1.175)

126 =; (6; 1 1) ; (10; 3 1); ; (10 1 3) ; (15; 2 2) (1.176) under the Pati-Salam group and thus the ﬁelds which can develop a SM-invariant VEV are (10 3 1), (10 1 3), (1 2 2) and (15 2 2). With the exception of the last one we already encountered these representations in the context of the Pati-Salam model vL ; ; ; vR ; ; ; (cf. Sect. 1.3.2). Let us also≡ ﬁx the following notation≡h for the SM-invarianti VEVs vu;d ; ; u;d126 ; vu;d ; ; 126u;d ; ≡ (10 3 1) ≡ (10 1 3) (1.177) D E D E 10 10 126 126 (1 2 2) (15 2 2) (1.178)≡ √ 26 i 1 Alternatively one can imagine a complex 10 as the linear combination of two real 10’s, i.e. 10 1 2 2 (10 + 10 ). This should make clearer the origin of the new structures in Eq. (1.172). SO

Chapter 1. From the standard model to 54 vu;d ; ; u;d ; vu;d ; ; u;d : (10) ≡ ≡ 1 D E 15 D E 120 120 120 ; ; ;120; (1 2 2) (15 ⊕2 2) (1.179) Given the embedding of a SM fermion family into (4 2 1) (4 1 2) (c.f. Eq. (1.105)) one u u u u ﬁnds the following fermionMu Y massv sumY rulev afterY thev electrowev ak symmetry breaking d d d d Md Y v Y v Y v 1 v 15 10 10 126 126 120 120 120 = d + d + ( d + d ) (1.180) Me Y v Y v Y v 1 v15 10 10 126 126 120 120 120 = u −+ u + ( u +− u) (1.181) MD Y v Y v Y v v 10 10 − 126 126 120 1201 − 12015 MR = Y vR 3 + ( 3 ) (1.182) 10 10 126 126 120 1201 12015 ML = Y vL 3 + ( 3 ) (1.183) 126 = (1.184) 126 MD MR ML = (1.185) ν;νc where , and enter the neutrinoML massMD matrix deﬁned on the symmetric basis T : ( ) MD MR (1.186) ; ; Me MD − h i 27 Eqs. (1.180)–(1.185) follow from the SM decomposition , but it is maybe worth of a comment the 3 factor in front of (15 2 2) for the leptonic components and . ; ; ; ; ; ; : That is understood by looking at the Pati-Salamh i invariant

SU C SU C (4 2 1) (15 2 2) (4 1 2) (1.187)⊗ U Q ; ; h i The adjoint of (4) is a traceless hermitian matrix, so the requirement of an (3) v (1) preserving vacuum; implies; the following; ; ; shape for (15u 2; 2) h i ∝ − ⊗ vd 0 (15 2 2) diag(1 1 1 3) (1.188) 0 ; ; − h i which leads to an extra 3 factor for leptons with respect to quarks. Conversely (1 2 2) preserves the symmetry between quarks and leptons.vR In order to understandSO the implications of the sum-rule in Eqs. (1.180)–(1.185) it is useful to estimate the magnitudeMU of the VEVs appearing there: is responsible for the rank reductionv of (10) and gauge unification constrains its value to be aroundvL (or just M /MU below)O W the unification scale , then all the bi-doublets can develop a VEV (collectively denoted2 as ) which is at most of the order of the28 electroweak scale, while is a small ( ) VEV induced by the scalar potential in analogy to what happens in the left-right symmetric models (cf. Sect. 1.3.1). 27 SO 28For a formal proof see e.g. [108]. In the contest of (10) this was pointed out for the first time in Ref. [33]. SO

1.5. Yukawa sector in renormalizable (10) 55 vR v vL ≫ ≫ Thus, given the hierarchy , Eq. (1.186) can be block-diagonalized T Mν ML MDM− M ; (cf. Eq. (1.99)) and the light neutrino mass− matrixR is veryD well approximated by 1 = (1.189) where the ﬁrst and the second term are the type-II and type-I seesaw contributions already encountered in Sect. 1.3.1. Which is the minimum number of Higgs representations needed in the Yukawa sector in order toF have a realistic theory? With only one Higgs representation at play there is no fermion mixing, since one Yukawa matrix can be always diagonalized by rotating the 16 ﬁelds, so at least two of them must be present. Out of the six H H combinations⊕ (see e.g. [116]):

H H 1. 10 ⊕126

H H 2. 120 ⊕ 126

H H 3. 10 ⊕ 120

H H 4. 10 ⊕10

H H 5. 120 ⊕ 120 6. 126 126 Md Me Md Me Y − them last three can be readily discardedm sincem they predict wrong mass relations, namely − 120 = (case 4), H = 3 (case 6), while in case 5 the antisymmetry of implies 1 2 3 = 0 (first generation) and = (second and third generation). Notice that in absenceH of H126 (case 3) neutrinos are Dirac andH their mass is related to that of charged leptons which is clearly wrong. In order to cure this one has to introduce the bilinear 16 16 which plays effectively the role of 126 (cf. Sect. 4.1 for a discussion of this case in the context of the Witten mechanism [67, 117, 118]). Though all the cases 1, 2 and 3 give rise to well defined Yukawa sectors, for definiteness we are going to analyze in more detail just the first one. 1.5.1 H H with supersymmetry ⊕

10 126 H H H H ⊕ ⊕ ⊕ This case has been the most studied especially in the context of the minimal supersymmetric version, featuring 210 126 126 10 in the Higgs sector [46, 47, 48]. The eﬀective mass sum-rule in Eqs. (1.180)–(1.185) can be rewritten in the following SO

Chapter 1. From the standard model to 56 (10)

M Y v Y v ; way u u u M Y v10 Y v126 ; d 10 d 126 d = + Me Y v10 Y v126 ; 10 d − 126 d M = Y v10 + Y v126 ; D 10 u 126 u = − 3 (1.190) MR Y v10R ; 126 10 126 ML = Y vL ; 3 126 = Y 126Y =

Mν 10 ML 126MDM− MD : and, exploiting the symmetry of and − , theR neutrino mass matrix reads 1 = (1.191) 29 In the recent years this model received a lot of attention due to the observation [134] b τ that the dominance of type-II seesaw leads to− a nice correlation between the large atmospheric mixing in the leptonic− sector and the convergence of the bottom-quark θ ◦ and tau-lepton masses at the unification scale ( unification)∼ which is a phenomenon occurring in the MSSM up to 20 30% corrections [86]. b τ 13 Another interesting prediction− of the model is 10 [120], in agreement with the recent data released by the T2K collaboration [135]. Mν ML The correlation between unification and large atmospheric mixing can be Mν Md Me : understood with a simple two generations∝ − argument. Let us assume = in Eq. (1.191), then we get (1.192) In the the basis in which charged leptons are diagonal and for small down quark ms mµ mixing , Eq. (1.192) is approximatedMν by− ; ∝ mb mτ − (1.193) mb mτ and, being the 22 entry the largest one, maximal atmospheric mixingY requiresY a cancellation between Mande . Md Mu;MD 10 126 ForMν a more accurate analysis [53] it is convenient to express the and Yukawa matrices in terms of and , and substitute them in the expressions for Mu fu r Md r Me ; and : −

MD fu r Md r M e ; = (3 +− ) + (1 ) (1.194) = 3(1 ) +(1+3 ) (1.195) 29 For a set of references on the subject see [119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133]. SO

1.5. Yukawa sector in renormalizable (10) 57 v v v f u ; r d u : u v where vd10 u10 vd126 1 = 10 = 10 126 (1.196) 4 MD MD The neutrino mass matrixMν fν is obtainedMd Me as ξ Md Me − ; − fu − fu 1 = ( ) + ( ) (1.197)

vL fuvd fν ; ξ : v v126 2 with vd − L R 1 4 = 126 = mx x u;d;e;ν (1.198) 4 In what follows we denote diagonal mass matrices byMd ˆ ,md= , withM eigen-e values corresponding to the particle masses, i.e. being reaMl ande W positive.e†meWe∗ We chooseWe a basis where the down-quark matrix is diagonal: fu = ˆfν. In this basis is a general complex symmetric matrix, that can be written as = ˆ , where is a general unitary matrix. Without loss of generalityWe andfu; fν can be taken to be real and positive. Hence,r theξ independent parameters are given by 3 down-quark masses, 3 charged lepton masses, 3 angles and 6 phases in , , together with two complex parameters and : 21 real parameters in total, among which 8 phases. Using Eqs. (1.194), (1.195), and (1.197) all observables (6 quark masses, 3 CKM angles, 1 CKM phase, 3 charged lepton masses, 2 neutrino mass-squared differences, the mass of the lightest neutrino, and 3 PMNS angles, 19 quantities altogether) can be calculated in terms of these input parameters. Since we work in a basis where the down-quark mass matrix is diagonal the CKM T matrix is given by the unitary matrixmu diagonalizingWuMuWu the up-quark mass matrix up to diagonal phase matrices: iβ îβ = iβ iα iα (1.199) Wu e ;e ;e V e ;e ; ; with 1 2 3 1 2 αi;βi CKM = diag( ) T diag( 1) (1.200) mν WνMνWν T whereWe∗Wν areD V unobservableD phasesD atD low energy. The neutrino mass matrix given in Eq. (1.197) is diagonalized by ˆ = , and the PMNS matrix is determined 1 PMNS 2 1 2 by = ˆ ˆ , where ˆ and ˆ are diagonal phase matrices similar to those in Eq. (1.200). Allowing an arbitrary Higgs sector it is possible to obtain a good fit of the SM flavor structure [53]. However, after including the constraints of the vacuum in the minimal supersymmetric version of the theory [49, 50, 51], one finds [52, 53] an irreducible incompatibility between the fermion mass spectrum and the unification constraints. The reason can be traced back in the proximity between the unification scale and the seesaw scale, at odds with the lower bound on the neutrino mass scale implied by the oscillation phenomena. SO

Chapter 1. From the standard model to 58 (10)

The proposed ways out consist in invoking a split supersymmetry spectrum [136] or resorting to a non-minimal Higgs sector [137, 138, 139, 140], but they hardly pair the appeal of the minimal setting. In this respect it is interesting to notice that without H H supersymmetry⊕ gauge uniﬁcation exhibits naturally the required splitting between the seesaw and the GUT scales. This is one of the motivations behind the study of the 10 126 system in the absence of supersymmetry. 1.5.2 H H without supersymmetry ⊕ H 10 126 mt mb In the nonsupersymmetric case it would be natural to start with a∼ real 10 . However, as pointed out in Ref. [141] (see also [142] for an earlier reference), this option is not phenomenologicallyθq V viable.cb The reason is that one predicts , at list when working in the two heaviest generations limit with real parameter and in the sensible T approximation = = 0. It is instructiveWu to reproduce this statementWu with the help of the parametrization given in Sect. 1.5.1. Md md

Let us start from Eq. (1.194) and apply T (from the left) andT (from the right). mu fu r mdWuW r WuMℓW : Then, taking into account Eq. (1.199) and theu choice− of basis u = ˆ , we get ˆ = (3 + ) ˆ + (1 ) (1.201) T WuW Next• we makeu ∼ the following approximations:

Wu V • ∼ 1 (real approximation)

V CKM Vcb Vts Aλ • ∼ (real approximation) ∼ ∼ ∼ T 2 WCKMuMℓW mℓ • 1u (for∼ the 2nd and 3th generation and in the limit 0) ˆ (for the self-consistency of Eq. (1.201) in the limits above)

mc fu r ms r mµ ; which lead to the system ∼ − mt fu r mb r mτ : ∼ (3 + ) + (1 − ) (1.202) r (3 + ) + (1 ) (1.203)

mc mτ mb mt mµ ms mt It is then a simple algebrafu to substitute− back− and− ﬁnd the relation: ∼ msmτ mµmb ∼ mb − 1 ( ) ( ) 1 u d H v v (1.204)fu 4 | | | | 4 10 10 On1 the other hand a real 10 predicts fu = : and hence from Eq. (1.196) = 4 × ∼ O . More quantitatively, considering16 the nonsupersymmetric running for the fermion masses evaluated at 2 10 [143], one gets 22 4, which is oﬀ by a factor of (100). SO

1.5. Yukawa sector in renormalizable (10) 59 H

This brief excursus shows that the 10 must be complex. In such a case the fermion M Y v Y v Y v ; mass sum-rule reads u u d ∗ u 10 10 126 Md Y vd Y vu ∗ Y vd ; 10 ˜10 126 = 10 + 10 + 126 Me Y vd Y vu ∗ Y vd ; 10 ˜10 − 126 = 10 + 10 + 126 MD Y10vu Y10vd ∗ Y126vu ; = + ˜ − 3 (1.205) MR Y v10R ; 10 126 10 ˜10 126 ML = Y vL :+ 3 126 = 126 =

The three different Yukawa sourcesY would certainly weaken the predictive power of the model. So the proposal in Ref. [141] was to impose a Peccei-QuiPQ Fnn (PQ)α PQ symmetryH [144,α 10 − PQ H α ˜ 145] which forbids− the coupling , thus mimicking a supersymmetric Yukawa sector (see also Ref. [142]).H The following charge assignment:U R U B (16L ) =U ,Y (10 ) =U 2PQ ⊗ − Ï SO U PQ and (126 ) = 2 would suffice. ⊗ In this case 126 is responsibleU PQ bothU Y for (1) Y (1) (1) and the (1) ⊗ ⊥ ⊥ breaking.Y However, since it cannot break the rank of (10) (1) by two units, a global linear combination of (1) (1) (where is the generator orthogonal to ) survives at the electroweak scale. This remnant global symmetry is subsequently broken by the VEV of the electroweak doublets, that is phenomenological unacceptable ÷ B L since it would give rise to a visible axion [146, 147] which is experimentally− excluded. Actually9 astrophysical12 and cosmological limits prefers the PQ breaking scale in the window 10 GeV (see e.g. [148]). It is therefore intriguing to link the breaking scale responsible for neutrino masses and the PQ breaking one in the same model. This has been proposed long ago in [149] and advocatedU againPQ in [141]. What is needed is another representation charged under the PQ symmetry in such a way that it is decoupled from the SM fermions and which breaks (1) completely at very high scales. SOIn summary the PQ approach is very physical and well motivated since it does not just forbid a coupling in the Yukawa sector making it moreθ "predictive", but correlates (10) with other two relevant questions: it offers the axionSO30 as a dark matter candidate and it solves the strong CP problem predicting a zero . However one should neither discard pure minimal (10) solutions with the SM as the effective low-energy theory. Notice that in the PQ case we are in the presence of a 2HDM which is more than what required by the extended survival hypothesis (cf. the discussion in Sect. 1.2.4) in order to set the gauge hierarch31 y. Indeed two different

fine-tunings30 are needed in order to get two light doublets . 31This is trueHu as longHd as we ignore gravity [150]. The situation is different in supersymmetry where the minimal fine-tuning in the doublet sector makes both and light. SO

Chapter 1. From the standard model to 60 (10) vd vd vu vu 10 M126d ThusMe we could10 minimally126 consider the sum-rule in Eq. (1.205) with either = = 0 or = = 0. The ﬁrst option leads to a clearly wrong conclusion, i.e. M Y v ; = . So we are left with theu secondd one∗ which implies M Y v10 Y v ; d 10 d d = ˜ Me Y v10 Y v126 ; 10 d − 126 d = 10 + 126 MD Y10vd ∗ ; 126 = 3 (1.206) MR Y v10R ; ˜10 ML = Y vL ; 126 = 126 = Mν ML MDM− MD : − R 1 and MD Mu MR = Md Me (1.207) − Notice that in the case of type-I seesaw the strong hierarchy due to = must by undone by which remains proportional to . More explicitly, in the case of vd type-I seesaw, one ﬁnds Mν Mu Md Me − Mu : − v126R 1 =4 ( ) (1.208)

Though a simple two generations argument with real parameters shows that Eq. (1.208) could lead to an incompatibility with the data, a full preliminary three generations study 1.5.3indicates Type-I that this vs is not type-II the case seesaw [151].

Here we would like to comment about the interplay between type-I and type-II seesaw in b τ Eq. (1.189). In a supersymmetric context one generally expects these two contributions− to be comparable. As we have previously seen (see Sect. 1.5.1) the dominance of type-II b τ seesaw leads to a nice− connection between the large32 atmospheric mixing and mb : : mτ : : uniﬁcation and one would like to keep± this feature . On the other± hand without supersymmetry× the convergence is far from being obtained. For instance the running within the16 SM yields = 1 00 0 04 GeV and = 1685 58 0 19 MeV at the scale 2 10 GeV [143]. Thus in the nonsupersymmetric case the dominance of type-II seesaw would represent a serious issue. SO In this respect it is interesting to note that the type-II seesaw contribution can be naturally subdominant in nonsupersymmetricSO (10). The reason has to do with the32 left-right asymmetrization of the scalar spectrum in the presence of intermediate See e.g. Ref. [152] for a supersymmetric (10) model in which the type-II seesaw dominance can be realized. 1.6. Proton decay 61

SU R R H B L ⊂ 33 − symmetryMU breaking stages .M UsuallyR the uniﬁcation pattern is such that the mass of the (2) vtripletR ∆R 126 responsible for the breaking is well below the GUT ≡h i ∆ M M SU L L H MU scaleR . TheL reason is that must be ﬁne-tuned⊂ at the level of the34 intermediate scale VEV ∆ . Then, unless therevL is aL discrete left-right symmetry which locks ∆ ∆ ≡h i = , the mass of the (2) tripletv ∆ 126 , remains automatically at . v λ R v : On the other hand the induce VEV L ∆M is given by (cf. e.g. Eq. (1.103)) L 2 2 λ v = ∆ (1.209) where and denote a set of parameters of the scalar potential and an electroweak MB L v VEV respectively. Sov weR canMB writeL ; vL λ − ; ∼ − ∼ MU 2 MB2 L −

MB L/MU (1.210) − 2 which shows that type-II seesaw is suppressed by a factor ( ) with respect to 1.6type-I. Proton decay

The contributions to the proton decay can be classified according to the dimension of the baryond violating operators appearing in the lagrangian. Since the externald states ared fermions and because of the color structure the proton decay operators arise first at the = 6 level. Sometimes the source of the baryon violation isd hidden in a =5or a = 4 operatord involving also scalar fields. These operators are successively dressed with the exchange of other states in order to get effectively the = 6 ones. The so-called = 6 gauge contribution is the most important in nonsupersymmetric GUTs. In particular if the mass of thed lepto-quarks which mediate these operators is constrained by the running then the major uncertainty comes only from fermion mixing. There is also another class of = 6 operators coming fromd the Higgsd sector but they are less important and more model dependent. The supersymmetrization of the scalar spectrum gives rise to = 5 and = 4 baryon and lepton number violating operators which usually lead to a strong enhancement of the proton decay amplitudes, though they are very model dependent. In the next subsections we will analyze in more detail just the gauge contribution while we will briefly pass through all the other ones. We refer the reader to the reviews [154, 155, 156] for a more accurate account of the subject. 33 SO 34ForH a similarH phenomenon occurring in the context of left-right symmetric theories see Ref. [153]. As we will see in Chapter 2 this can be the case if the (10) symmetry breaking is due to either a 54 or a 210 . SO

Chapter 1. From the standard model to 1.6.162 (gauge) (10) d d = 6 SU C SU L U Y Following⊗ the approach⊗ of Ref. [157], we start by listing all the possible = 6 baryon number violating operators due to the exchange of a vector boson and invariant under B L c µ c (3) (2) (1)OI[25,− 158,k 159]ijk αβ uia γ qjαa eb γµ qkβb ; B L c µ c O − k2 u γ q d γ ℓ ; II 1 ijk αβ ia jαa kb µ βb B L = c µ c (1.211) O − k2 d γ q u γ ℓ ; III 1 ijk αβ ia jβa kb µ αb B L = c µ c (1.212) O − k2 d γ q ν γ q : IV 2 ijk αβ ia jβa b µ kαb = 2 (1.213) 2 k = gU /√ MX k gU /√ MY MX MY (1.214)MU ∼ gU 1 2 In the above expressions i;j;k= 2; ; and = 2SU , whereC α;β ;, SU L and a areb the masses of the superheavy gaugeq u;d bosons andℓ theν;e gaugeSU couplingL at the B L B L unification scale. The indicesOI − =1OII−2 3 are referred to (3) , =1 2 to (2) and and Xare family; ; indices. The fields = ( ) and = ( ) are (2) doublets.SU − B L B L The effectiveY operators; 5 ; and appear whenO weIII− integrateOIV− out the superheavy 6 gauge fieldSU = (3 2 ). This1 is the case inSO theories based on the gauge group X (5). IntegratingY out = (3 2 + 6) we obtain the operators and . This is the case of flipped (5) theories [69, 70], while in (10) models both the lepto-quarks and are present. Using the operators listed above, we can write the effective operators in the physical c c c µ c basis for each decayO e channelα;dβ c [157]eα;dβ ijk ui γ uj eα γµ dkβ ; c c c µ c O eα;dβ c eα;dβ ijk ui γ uj dkβ γµ eα ; ( ) =c ( ) c c µ c (1.215) O νl;dα;dβ c νl;dα;dβ ijk ui γ djα dkβ γµ νl ; ( c ) =c ( c ) c c µ c (1.216) O νl ;dα;dβ c νl ;dα;dβ ijk diβ γ uj νl γµ dkα ; ( ) = ( ) (1.217) ( ) = ( ) (1.218)

c αβ β α where c eα;dβ k V V V VUD V VUD† ; c βα β α c e ;d k2 V11V k V V1† V V 1 V †V ; α β 1 1 2 1 UD 2 UD ( ) =c [ + ( α ) (βl ) ] βα l (1.219) 2 11 2 1 1 † c νl;dα;dβ 1 k 1 V3 VUD 2 V V4 EN k1 V 4V3VUDV V VEN ; ( c ) =c + β ( lα) ( βα ) l (1.220) 2 † 1 † 2 † † 1 c νl ;dα;dβ k1 V1 VUD U3EN V V2 U4 EN V1 VUD 4 3; ( ) = 2 ( ) 1( ) + ( 1 ) (1.221) 2 4 2 4 2 α β( ) = [( ) ( ) + ( ) ] (1.222) 6 C C V Uc†U V Ec†D V Dc†E V Dc†D VUD U†D VEN E†N UEN E †N with = = 2. In the equations above we have deﬁned the fermion mixing matrices as: 1 2 3 4 = , = , = , = , = , = and = , 1.6. Proton decay U;D;E 63

UT Y U Y diag ; where define the Yukawa couplingc U diagonalizationU so that T diag Dc YD D YD ; T = diag (1.223) Ec YE E YE ; T = diag (1.224) N YN N YN : = (1.225) VUD U†D K VCKM K K K = D (1.226) VEN K VPMNS K M1 2 1 2 Further, one may write =VEN K=VPMNS , where and are diagonal 3 4 matrices containing respectively three and two phases. Similarly, = in 3 the case of Dirac neutrinos,d or = in the Majorana case. k k b V FromV V thisV briefU excursusEN we can see that the theoretical predictions of the proton 1 2 lifetime1 from the gauge = 6 operators require the knowledge of the quantities , , 1 2 3 4 d B L , , , and . In addition we have three− (four) diagonal matrices containing phases in the case of Majorana (Dirac) neutrino. Since the gauge = 6 operators conserve the nucleon decays into a meson and an antilepton. Let us write the decay rates for the different channels. We assume that in the proton decay experiments one can not distinguish the flavor of the neutrino andd the chirality of charged leptons in the exit channel. Using the Chiral Lagrangian techniques (see e.g. [160]), the decay rates of the different channels due to the gauge p K ν = 6→ operators are [157] + mp mK mp c mp c Γ( ) − AL α D c νi;d;s D F c νi;s;d ; πm f2 | | 3 mB mB 2 2 p π 2 i ( ) 2 2 X 2 = 3 2 ( )+[1+ ( + 3 )] ( ) (1.227) 8 =1 3 3 mp c p π ν AL α D F c νi;d;d ; → πf | | 3 | | π i + 2 2 2 X 2 2 Γ( )= mp mη (1 + + ) ( ) (1.228) =1 c c p η eβ 8 − AL α D F c eβ;d c eβ;d ; → 2πfπm2p 2 | | − | | + 2 2 2 n 2 2o ( m m) Γ( )= p 2 3K (1 + mp3 ) ( ) +c ( ) c (1.229) p K eβ 48 − AL α D F c eβ;s c eβ;s ; → 2πfπ m2p 2 | | mB − | | 0 + ( m ) 2 2 2 n 2 2o p 2 3 c c Γ(p π eβ )= AL α D[1+F (c eβ;d)] (c eβ;d) + ; ( ) (1.230) Ï 8πfπ | | | | 0 + 2 2 2 n 2 2o 2 Γ( νi νe)=;νµ;ντ eβ (1 +e; µ+ ) ( ) + ( ) mB : (1.231) 16 ∼ mB m m fπ ∼ ∼ ∼ Dwhere: = F :and = . In the equations above 1 15 MeV is the ∼ ∼ Σ Λ average baryon mass , 131 MeV is the pion decay constant,α : ∼ − 0 80 and 0 47 are low-energy constants of the Chiral Lagrangian which can be obtained3 from the analysis of semileptonic hyperon decays [161] and 0 0112 GeV is a proton-to-vacuum matrix element parameter extracted via Lattice SO

Chapter 1. From the standard model to 64 (10) AL : ∼ MZ QCD techniques [162]. Finally 1 4 takes into account the renormalization from to 1 GeV. SO H H In spite of the complexity and the model-dependency of the⊕ branching ratios in Eqs. (1.227)– (1.231) the situation becomes much more constrained in the presence ofU symmetricc UKu Yukawas,Dc DKd relevantEc forEK realistice K(10)u Kd modelsKe based on 10 126 in the Yukawa sector. In that case we get the following relations for the mixing matrices: = ,

= and = , where , and / are diagonal matrices involving phases. These relations lead to the remarkable predictionQ [157] k / ; A V 1 A4 V CKM 1 CKM 1 | | | | | | | | 2 11 2 2 12 2 1 4 = 1 2 (1.232) + πm f p K ν m m whereQ p π Ï ; A p D ; A p D F : m 3 2m A α+ mB mB p − K L| | 1 8 Γ( ) 1 2 2 = 2 2 2 2 2 = = 1 + ( +3 ) (1.233) ( ) k gU /√3 MX 3 gU MX 1 Notice that the expression for = 2 is independent from unknown mixing matrices andSO CP violating phases, whileH theH values of and are subject to gauge p K ν ⊕ coupling unificationÏ constraints. This is a clear example of how to test a (nonsupersymmetric) (10)+ model with 10 126 in the Yukawa sector through the decay MX MY MU channel Γ( ). ∼ ∼ We close this subsection with a naive model-independent estimate for the mass of the superheavy gauge bosons . Approximating the inverse lifetime of m the proton in the following way (cf. the real computationp in Eqs. (1.227)–(1.231)) p αU ∼ MU5 2 4 τ p π e > : Γ (1.234) → × 0 + 33 MU > : ; and using ( ) 8 2 10 yr [11],× one finds the naive lower bound 15 αU− 2 3MU10 GeV αU− (1.235) 1 1 1.6.2where we fixed (scalar)= 40. The bound on as a function of is plotted in Fig. 1.3. d

= 6 d T ; ; In nonsupersymmetric scenarios the next-to-leading− contribution to the decay of the proton comesSU from the Higgs induced = 61 operators. In this case the proton decay is mediated by scalar leptoquarks = (3 1 3). For deﬁniteness let us illustrate the case of minimal (5) with just one scalar leptoquark. In this model the scalar leptoquark 1.6. Proton decay MU 65 4.0 ´ 1015 3.5 ´ 1015 3.0 ´ 1015 2.5 ´ 1015 2.0 ´ 1015 1.5 ´ 1015 Α -1 10 20 30 40 50 60 70 80 U

MU αU− 1 Figure 1.3: Naive lower bound on the superheavy gauge boson mass as a function of .

H lives in the 5 representation together with the SM Higgs. The relevant interactions for proton decay can be written in the following way [155]

T cT c T ijk αβ q C A qjβ Tk u C B e Ti L iα i T cT c αβ qiα C C ℓβ Ti∗ ijk ui C D dj Ti∗ h:c: = + + + + (1.236)

A B C D In the above equation we have used the same notation as in the previous subsection. T The matricesSU, , and are linear combinationsA ofB theY YukawaU C couplingsD YD inY theE theory and the possible contributions coming from higher-dd imensional operators. In the minimal (5) we have the following relations: = = , and = = = . Now, using the above interactions we can write the Higgs = 6 eﬀective operators for proton decay [155]

T T OH dα;eβ a dα;eβ u LCdα u L Ceβ ; c c T c c OH dα;eβ a dα;eβ u LCdα eβ† L Cu ∗ ; ( c ) = ( c ) c c T (1.237) OH dα;eβ a dα;eβ dα† LCu ∗ u L Ceβ ; ( c c) = ( c c) c c c c (1.238) OH dα;eβ a dα;eβ dα† LCu ∗ eβ† L C− u ∗ ; ( ) = ( ) T T 1 (1.239) OH dα;dβ;νi a dα;dβ;νi u LCdα dβ LCνi ; ( c) = ( ) c c c T (1.240) OH dα;dβ;νi a dα;dβ;νi dβ† LCu ∗ dα L C− νi ; ( ) = ( ) 1 (1.241) ( ) = ( ) (1.242) SO

Chapter 1. From the standard model to 66 (10)

T T T a dα;eβ U A A D α U CE β ; where MT c 12 T T 1 1 a(dα;eβ) = (U (A + A )D) α (Ec†B†U)c∗ β ; (1.243) MT c 12 T1 1 a(dα;eβ) = (Dc†D( †U+c∗ α )U) C(E β ; ) (1.244) MT c c 12 1 1 a(dα;eβ) = (Dc†D†Uc∗)α (Ec†B†U)c∗ β ; (1.245) MT 12 T 1 T 1T a(dα;dβ);ν =i ( U A ) A( D α )D CN βi ; (1.246) MT c 12 1T a(dα;dβ;νi) = ( Dc†(D†+Uc∗ β ) D) C(N αi : ) (1.247) MT 1 1 L γ / MT 2 α β ; SU L i ; ; − ( ) = ( ) ( ) (1.248) SU C 5 Here = (1 ) 2, is the triplet mass, = =1d 2 are (2) indices and =1 2 3 are (3) indices. The above analysis exhibits that the Higgs = 6 contributions are quite model A A D i j dependent,ij − ji and becauseij of this it is possible to suppress them in specific models of fermion masses. For instance, we can set to zero these contributions by the constraints = and = 0, except for = = 3. Also in this case we can make a naive model-independent estimation for the mass of the scalar leptoquark using the experimentalm lowerp bound on the proton lifetime. Approximating the inverse lifetime ofp theYu protonYd in the following way ∼| | MT5 2 τ p π e > : 4 → × Γ (1.249) 0 + 33 MT > : : × and taking ( ) 8 2 10 yr [11], we11 find the naive lower bound 4 5 10 GeV (1.250) This bound tells us that the triplet Higgs has to be heavy, unless some special condition on the matrices in Eq. (1.236) is fulfilled (see e.g. [84, 85]). Therefore since the triplet Higgs lives with the SM35 Higgs in the same multiplet we have to look for a doublet-triplet 1.6.3splitting mechanism . d d = 5 q q q ℓ uc uc dc ec In the presence of supersymmetryMT new = 5 operatorsMT of the type 1 1 ˜ ˜ and ˜ ˜ (1.251) 35 Cf. Sect. 4.4.3 for a short overview of the mechanisms proposed so far. 1.6. Proton decay 67 MT

c c c c are generated via colored triplet Higgsino exchangeqqqℓ withu u madsse [159, 163]. These operators can be subsequently dressed at one-loop with an elMectroweakT m gauginom (gluino or wino) or higgsino leading to the standard and operators. Since the amplituded turns out to be suppressed just by the product ˜ , where ˜ is the soft d p scale, this implies a generic enhancement of the proton decay rate with respect to theÏ Kordinaryνµ = 6 operators. p π e Ï c c c c + Another peculiarity of = 5 operators is that the dominantqiqj qkℓl decayui u0 modej d+kel is i;j;k;lwhich; differs; from the standard nonsupersymmetric mode . A simple symmetry argumentSU C showsSU the reason:L the operators ˆ ˆ ˆ ˆ and ˆ ˆ ˆ ˆ (where = 1 2 3 are family indices and color and weak indices are implicit) must be invariant under (3) and (2) . This means that their color and weak indices must be antisymmetrized.i j Howeverk since this operators arei givenj by bosonic superfields, they must be totally symmetric under interchange of all indices. Thus the first operator vanishes for = = and theSU second vanishes for = . Hence a second or third generation memberqqqℓ must exist in the final state. In minimal supersymmetric (5) [94] the coefficient of the baryon number violat- α Y Y mg ing operator can be schematically written as; (see e.g. [164]) π MT m 3 10 5 q˜ 2 ˜ (1.252) 4 mq mg Y Y where we have assumed the dominance of the gluino exchange and that the sfermion ˜ ˜ 10 5 masses ( ) are bigger than the gluino one ( ), while and areSU couplings of the Yukawa superpotential. Though there could be a huge enhancement of the proton decay rate which brought to the claim that minimal supersymmetric (5) was ruled out [165, 166], a closer look to the uncertainties at play makes this claim much more weaker• [167]: SU T The Yukawa couplings in Eq. (1.252) areMd not directlyMe related to those of the SM, since in minimal (5) one needs to take into account non-renormalizable operators in order to break the relation = , and thus they can conspire to • suppress the decay mode [168]. A similar suppression could also originate from the soft sector even after includ- MT • ing the constraints coming from flavor violating effects [169]. Last but not least the mass of the triplet is constrained by the running only in the renormalizable version of the theory [166]. As soon as non-renormalizable operators (which are anyway needed for fermion mass relations) are included this is not true anymore [167]. In this respect it is remarkable that even in the worseMT case scenario of the renormalizable theory the recent accurate three-loop analysis in Ref. [170] increases by about one order of magnitude the upper bound on due to the running constraints. SO

Chapter 1. From the standard model to 68 SU (10)

Thus the bottom-line is that minimal supsersymmetric (5) is still a viable theory and more input on the experimental side is needed in order to say something accurate on 1.6.4proton decay. d R = 4 c ′ c ′′ c c c This last class of operatorsWRPV µi originatesℓihu λijk fromℓiℓiei theλijk -parityqiℓj dk violatingλijk ui dj superpotentialdk : of the MSSM λ′′ µ λ λ ˆ ˆ ˆ ˆ ˆ ˆ ′ ˆ ˆ = + ˆ + ˆR + ˆ (1.253)

Notice that violates baryon number′ whilec ′′ , candc c violate lepton number. So for RPV λ qiℓj d λ u d d : instance we have the followingL ⊃ interactionsijk k inijk thei j -parityk violating lagrangian c c c d ˜ + ˜ + h.c. qℓu †d † (1.254)

The tree-level exchange of ˜ generates the baryon violating operator with a λ′ λ′′ /m : coeﬃcient which can be written schematicallydc as 2 ˜ (1.255)

Barring cancellations in the family structure of this coefficient and assuming a TeV λ′ λ′′ . − : scale soft spectrum, the proton lifetime implies the generic bound [171] 26 R 10 (1.256) SU It’s easy to see that the -parity violating operators are generated in supersymmetric GUTs unless special conditions are fulfilled. For instance in (5) the effective trilinear couplings originate from the operator ijk i j k ;

λ λ′ λ′′ SO R Λ 5ˆ 5ˆ 10ˆ (1.257) 1 2 ijk which leads to = = = Λ. Analogouslyi j k in H (10): the -parity violating trilinears stem from the operator MP Λ D E 16ˆ 16ˆ 16ˆ 16ˆ (1.258) Z If one doesn’t like small numbers such asSO in Eq. (1.256) the standard approach is 2 to impose a matter parity whichSO forbids the baryon andR lepton number violating operators [94]. A more physical option in (10) is instead suggested by Eq.R (1.258). Actually it seems that as soon as (10) is preserved the -parity violating trilinears are not generated. In order to better understandB L thisS point let us rephrase the -parity RP − ; in the following language [172] − 3( )+2 = ( ) (1.259) 1.6. Proton decay S 69 B L SO − where the spin quantum number is irrelevantB L as long;::: as the Lorentz symmetry is B L ; − ; ; ;::: preserved. Then, since is− a local generator of (10), it is enough to embed the SMR fermions in representationsSO with odd (e.g. 16 )R and the Higgs doublets B L H H in representations with even (e.g.− 10 120 126 210 ) in order to ensure exact -parity conservation. AfterSO the (10) breaking theRP fate of -parity depends on the order parameter responsible for the breaking. Employing either a 16 or a 126 RP H H RP H H : for the rank reductionh ofi −h(10) thei action of the h operatori h oni their VEV is R 16 = 16 or 126 = 126 (1.260) SO H In the latter case the -parity is preserved by theH vacuum and becomes an exact symmetry of the MSSM. This feature makes supersymmetricMB L/MP (10) modelsH withM 126B L − h i ∼ − very appealing [62]. On the other hand with a 16 atSO play the amount of R-parity B L violation− is dynamically controlled by the parameter , where 16 . Though conceptually interesting it is fair to say that in (10) it is unnatural to have the breaking scale much below the uniﬁcation36 scale both from the point of view of uniﬁcation constraints and neutrino masses .

H H MB L 36 − MB L As we will see in Chapter 2 when the GUT breaking is driven either by a 45 or a 210− there are vacuum configurations such that can be pulled down till to the TeV scale without conflicting withSO unification constraints. On the other hand the issue of neutrino masses with a low is more serious. One has either to invoke a strong fine-tuning in the Yukawa sector or extend the theory with an (10) singlet (see e.g. [173]). SO

Chapter 1. From the standard model to 70 (10) Chapter 2

Intermediate scales in nonsupersymmetric SO uniﬁcation

(10) SO The purpose of this chapter is to review the constraints enforced by gauge unification on the intermediate mass scales in the nonsupersymmetric (10) GUTs, a needed preliminary step for assessing the structure of the multitude of the different breaking patterns before entering the details of a specific model. Eventually, our goal is to envisage and examine scenarios potentially relevant for the understandingSO of the low energy matter spectrum. In particular those setups that, albeit nonsupersymmetric, may exhibit a predictivity comparable to that of the minimal supersymmetric (10) [46, 47, 48], BscrutinizedL at length in the last few years. MU − The constraints imposed by the absolute neutrinoSO mass scale on the position of the threshold, together with the proton decay bound on the unification scale , provide a discriminating tool among theSO many (10) scenarios and the corresponding breaking patterns. These were studied at length in the 80’s and early 90’s, and detailed surveys of two- and three-stepSO (10) breaking chains (one and two intermediate thresholds respectively) are found in Refs. [174, 100, 175, 64]. We perform a systematicU survey of (10) unification with two intermediate stages. In addition to updating the analysis to present day data, this reappraisal is motivated by (a) the absence of (1) mixing in previous studies, both at one- and two-loops in the gauge coupling renormalization, (b) the need for additional Higgs multiplets at some intermediate stages, and (c) a reassessment of the two-loop beta coefficients reported in the literature. The outcome of our study is the emergence of sizeably different features in some of the breaking patterns as compared to the existing results. This allows us to rescue previously excluded scenarios. All that before considering the effects of threshold corrections [176, 177, 178], that are unambiguously assessed only when the details of a specific model are worked out. Eventually we will comment on the impact of threshold effects in the Outlook of the thesis. SO

Chapter 2. Intermediate scales in nonsupersymmetric unification 72 SO(10) H H H H It is remarkable that the chains corresponding to the minimal (10) setup with the smallest Higgs representations (10 , 45 and 16 , or 126 in the renormalizable case) and the smallestSO number of parameters in the Higgs potential, are still viable. The complexity of this nonsupersymmetric scenario is comparable to that of the minimal supersymmetric (10) model, what makes it worth of detailed consideration. In Sect. 2.1 we set the framework of the analysis. Sect. 2.2 provides a collection of theβ tools needed for a two-loop study of grand uniﬁcation. The results of the numerical study are reported and scrutinized in Sect. 2.3. Finally, the relevant one- and two-loop -coeﬃcients are detailed in Appendix A. 2.1 Three-step SO breaking chains

SO G G SM →G →(10)G → The relevant (10) 2 1 symmetry breaking chains with two intermediate gauge groups 2 and 1 are listed in Table 2.1. Effective two-step chains are obtained by joining two of the high-energy scales, paying attention to the possible deviations from minimality of the scalar content in the remaining intermediate stageP (this we shall discuss in Sect. 2.3.2). For the purpose of comparison we follow closely the notation of Ref. [64], where denotes the unbroken D-parity [96, 97, 98, 99, 100]. For eachG step the Higgs representation responsible for the breaking is given. H H The breakdown of the lower intermediateH symmetry 1 to the SM gauge group is drivenSO either by the 16- or 126-dimensional Higgs multiplets 16 or 126 . An important feature of the scenariosH with 126 is the fact that in such a case a potentially realistic (10) Yukawa sector can be constructed already at the renormalizable level (cf. Sect. 1.5).F TogetherF withH 10 all theH effectiveH / Dirac Yukawa couplings as well asM theU Majorana mass matrices at the SM level emerge from the contractions of the matter bilinears 16 16 with 126 or with 16 16 Λ, where Λ denotes the scale (above ) at which the effective dimension five Yukawa couplings arise. D-parity is a discrete symmetry acting as chargeSO conjugation in a left-right symmetric context [96, 97], and as that it playsSO the role of a left-right symmetry (it enforces for instance equal left and right gauge couplings). (10) invariance then implies exact D- parity (because D belongs to the (10) Lie algebra). D-paritySU R may be spontaneously broken by D-odd Pati-Salam (PS) singlets contained in 210 or 45 Higgs representations. Its breaking can therefore be decoupled from the (2) breaking, allowing for different left and right gauge couplings [98, 99]. The possibility of decoupling the D-paritygL g breakingR from the scale of right-handed interactions is a cosmologically relevant issue. On the one hand baryon asymmetry cannot arise in a left-right symmetric ( = ) universe [96]. On the other hand, the spontaneous breaking of a discrete symmetry, such as D-parity, creates domain walls that, if massive enough (i.e. for intermediate mass scales) do not disappear, overclosing SO

2.1. Three-step breaking chains (10) 73

C L R C L R B L ÊÏ { } ÊÏ { − } Chain G2 G1 C L RP 45 C L R B LP I: ÊÏ210 {4 2 2 } ÊÏΛ {3 2 2 1 − } C L RP 210 C L R B L II: ÊÏ54 {4 2 2 } ÊÏΛ {3 2 2 1 − } C L R B LP 45 C L R B L III: ÊÏ54 {4 2 2 − } ÊÏΛ {3 2 2 1 − } C L R 45 C L R IV: ÊÏ210 {3 2 2 }1 ÊÏΛR {3 2 2 }1 C L RP 45 C L R 210 Σ V: ÊÏ {4 2 2 } ÊÏR {4 2 1 } C L RP 45 C L R VI: ÊÏ54 {4 2 2 } ÊÏλΣ {4 2 1 }

C L R B L 210 C L R B L − − VII: ÊÏ54 {4 2 2 } ÊÏR {4 2 2 } C L R B LP 45 C L R B L 45 − Σ − VIII: ÊÏ {3 2 2 1 } ÊÏR {3 2 1 1 } IX: C L R 45 C L R B L 210 Σ − ÊÏ {3 2 2 }1 ÊÏσR {3 2 1 1 } X: C L RP 210 C L R B L 210 − ÊÏ {4 2 2 } ÊÏσR {3 2 1 1 } XI: C L R 210 C L R B L ÊÏ54 {4 2 2 } ÊÏ {3 2 1 1 − } XII: 45 SO 45 4 2 1 Λ 3 2 1 1

T Relevant (10) symmetry breaking chains via two intermediate gauge groups G1 and SOable 2.1: C L Y G2. For each step the representation of the Higgs multiplet responsible for the breaking is given in (10) or intermediate symmetry group notation (cf. Table 2.2). The breaking to the SM group 3 2 1 is obtained via a 16 or 126 Higgs representation. the universe [97]. These potential problems may be overcome either by conﬁning D- parity at the GUT scale or by invoking inﬂation.SO The latter solution implies that domain walls are formed above the12 reheating temperature, enforcing a lower bound on the D-parity breaking scale of 10 GeV. Realistic (10) breaking patterns must therefore 2.1.1include this The constraint. extended survival hypothesis

every stage of at theThroughout symmetry all breaking three stages chain of onlyrunning those we scalars assume are that present the sca tlarhat spectrum develop a obeys vacuum the expectationso called extended value (VEV)survival at hypothesis the current (ESH) or the [82] subsequent which requ leviresels that of the spontaneous symmetry breaking

. ESH is equivalent to the requirement of the minimal number of ﬁne-tunings to be imposed onto the scalar potential [83] so that all the symmetry breaking steps are performed at the desired scales. On the technical side one should identify all the Higgs multiplets needed by the SO

Chapter 2. Intermediate scales in nonsupersymmetric unification 74 SO (10) SO C L R C L R C L R B L C L R B L { } { } { − } { − } ;Surviving; Higgs; ; multiplets; in; ; (10) subgroups; ; ; φ (10) 4; 2; 1 1 4 ;2 ;2 3 ;2 ;2 ;1 3; 2; 1 11; NotationδR10 2 − 2 − 10 (1 2 + 1) (1; 2; 2) (1; 2; 2; 0)1 (1 2 +1 0)1 δL16 2 2 2 2 16 (4 ;1 ;+ ) (4 ;1 ;2) (1 ;1 ;2 ; 1) (1 ;1 ;+ ; ) φ 16 2 16; ; 1 (4 ;2 ;1) (1; 2; 1; + ) ; ; ;1 126R 2 − 2− 126 (15 2 + ) (15; 2; 2) (1; 2; 2; 0) (1 2 + 0) L126 126 (10; 1; 1) (10; 1; 3) (1 1 3 1) (1 1 1 1) ∆126 126 (10; 3; 1) (1 3 1 1) ∆ 45 45 (15 1 0) (15; 1; 1) ; ; ; Λ210R 210 (15; 1; 1) ; ; ; Λ L45 45 (1 ;1 ;3) (1 1 3 0)σ ΣR45 45 (1 ;3 ;1) (1 3 1 0)σ ΣL210 210 (15; 1; 3) λ210 210 (15 3 1) 210 SO 210 (1 1 1) φ

Scalar multiplets contributing to the running of126 the gaugeC L couplingsR B L forC aL givenR B L (10) Table 2.2: − − subgroup according to minimalφ ﬁne tuning.φ The survival ofU B (notL required by minimality) is needed√ / − by aB realisticL / leptonic mass spectrum,10 as discussed126 in the text (in the 3 2 2 1 and 3 2 1 1 stages − only one linear combination of and φ remains). The (1) charge is given, up to a factor 3 2, by ( ) 2 (the latter is reported in the table).126 For the naming of the Higgs multiplets weH followH the notation of Ref. [64] with the addition of . When the D-parity (P) is unbroken the particle content must be left-right symmetric. D-parity may be broken via P-odd Pati-Salam singlets in 45 or 210 .

H B L breaking pattern under consideration and− keep them according to the gauge symmetry down to the scale of their VEVs. This typically pulls down a large numberH of scalars in scenarios where 126 providesG the breakdown. On the other hand, one must take into account that the role of 126 is twofold: in addition to triggering the 1 breaking it plays a relevant role in the Yukawa sector where it providesH the necessary breaking of the down-quark/charged-lepton; ; mass degeneracy; (cf.; Eq. (1.190)). For this to work one needs a reasonably large admixture 10 of the 126 component in the effective electroweak doublets. Since (1 2 2) can mix 126 with (15 2 2) only below the Pati-Salam breaking scale, both fields must be present at the Pati-Salam level (otherwise the scalar doublet massC mL atrixR does not provide large enough components of both these multiplets inH the light Higg; ;s fields). Note that the same argument; ; applies also to the 4 2 1 intermediate1 stage when 2 126 oneC L mustR B retainL theC L doubletR B L component1 of 126 , namely (15 2 + ) , in order for it to − − 10 eventuallyφ admixφ with (1 2 + 2) in the light Higgs sector. On the other hand, at the 3 2 2101 and126 3 2 1 1 stages, the (minimal) survival of only one combination of the and scalar doublets (see Table 2.2) is compatible with the Yukawa sector constraints because the degeneracy between the quark and lepton spectra has already equations

2.2. Two-loop gauge renormalization group 75 been smeared-outSO by the Pati-Salam breakdown. In summary, potentially realistic renormalizableH YukawaH textures; in; settings with well-separated (10) and Pati-Salam breaking scalesC L callR for an additional fine; tuning; 10 in the Higgs sector. In; the; scenarios with; ; 126 , the 10 CbidoubletL R (1 2 2) , included 126 in Refs [174, 100, 175, 64], must1 be paired at the1 4 2 2 scale withX, XIan extraXII. (15 2 2) 10 126 scalarF bidoublet (or (1 2 + 2) with (15 2 + 2 ) at the 4 2 1 stage). This can affect the runningφ of the gauge couplings in chains I, II, III, V, VI, VII, and or the sake126 of comparison with previous studies [174, 100, 175, 64] we shall not include the multiplets in the first part of the analysis. Rather, we shall comment 2.2on their Two-loop relevance for gauge gauge unification renormalization in Sect. 2.3.3. group equations

U ::: U N G ::: GN Gi In this section we⊗ report,⊗ in order⊗ to⊗ ﬁx⊗ a consistent′ notation, the two-loop renormalization group equations (RGEs) for the gauge couplings. We consider a gauge group 1 1 of2.2.1 the form The non-abelian(1) (1) sector , where are simple groups.

G ::: GN′ U ⊗ ⊗ 1 tLet us focusµ/µ first on the non-abelian sector corresponding to and defer dg the full treatment of the effects due to thep extra (1) factors to section 2.2.2. Defining 0 gp βp = log( ) we write dt p ;:::;N ′ = (2.1) β G ::: GN ⊗ ⊗ ′ where =1 is the gauge group label. Neglecting for the time being the abelian 1 components, the -functions for the gauge couplings read at two-loop gp level [179, 180,βp 181, 182, 183, 184]C Gp κS Fp ηS Sp π2 − 2 2 2 gp 2 11 4 1 = C (Gp) + C( F) +p C( )Gp κS Fp (4π2) − 3 3 3 2 34 2 2 20 2 2 + C2 S ( C( G)) +ηS 4S ( ) + ( ) ( ) (4 ) p 3 p p 3 g 2 2 2 2 + 4q ( ) + ( ) ( ) κ κC 3Fq S Fp ηC Sq S Sp Y Fp ; π2 − π ) h i 2 2 2 2 2 4 + 2 4 ( ) ( ) + ( ) ( ) 2 ( ) (2.2) κ ; (4 ) (4 ) η ; q p 1 6 1 where = 1 2 for Dirac and Weyl fermions respectively. Correspondingly, = 1 2 for complex and real scalar fields. The sum over = corresponding to contributions Chapter 2. Intermediate scales in nonsupersymmetric SO unification

76 (10) βp q F S R R ::: RN ⊗ ⊗ ′ to fromRp the other gauge sectors labelled by is understood.Gp Given a fermiond Rp 1 or a scalarS Rp ﬁeld that transforms according to the representation = , where is an irreducible representation of thed R group of dimension ( ), the 2 S Rp T Rp ; factor ( ) is deﬁned by ≡ d Rp 2 ( ) T Rp ( ) ( ) Rp (2.3) ( ) where ( ) is the Dynkin index of the representation . The corresponding Casimir eigenvalue is then given by C Rp d Rp T Rp d Gp ;

2 d G ( ) ( ) = ( ) ( ) (2.4) where ( ) is the dimension of the group. In Eq. (2.2) the ﬁrst row represents the one- loop contribution while the other terms stand for the two-loop corrections, including that induced by Yukawa interactions.Y Fp The latterC isF accountep Y Y † ; d for in terms of a factor d Gp 4 1 2 ( ) = Tr ( ) (2.5) ( ) abc where the “general” Yukawa couplingY ψaψb hc h:c:

+ (2.6) ψa;b hc includes family as well as group indices. The coupling in Eq. (2.6) is written in terms of four-component Weyl spinors and a scalar ﬁeld (be complex or real). The trace includes the sum over all relevant fermion and scalar ﬁelds. 2.2.2 The abelian couplings and mixing U

U (1) N µ In order to include theA abelianb contributions to Eq. (2.2)ψf at two loops and the one- and two-loop eﬀects of (1) mixing [185], letr us writeµ the most general interaction of abelian gauge bosons and a setψ off γ WeylµQf ψf fermionsgrbAb : as µ grb r;b ;:::;N Ab r r (2.7) Jµ ψf γµQf ψf N N grb × r U Qf The gauge couplingµ constants , = 1 , couple to the fermionic current = Ab. The gauge coupling matrixgrb can be diagonalized by two independentgrb r>b rotations: one acting on the (1) charges and theN N other/ on the gauge boson ﬁelds . For a given choice of the charges, can be set in a triangular form ( = 0 for ) by the gauge boson rotation. The resulting ( + 1) 2 entries are observable couplings. 2.2. Two-loop gauge renormalization group equations

a 77 Fµν

Since in the abelian case is itselfa aµν gauge invariant,a bµν the most general kinetic part F F ξabF F ; of the lagrangian reads at the− renormalizableµν − levelµν µ a b ξab < 1 1 A 6 | | a (2.8) 4 4 where = and 1. A non-orthogonal rotation of the fields may be performed to set the gauge kinetic term in a canonical diagonal form. Any further orthogonal rotation of the gauge fields will preserve this form. Then, the renormalization prescriptiongrb may be conveniently chosengrb to maintain at each scale the kinetic terms canonical and1 diagonal on-shell while renormalizing accordingly the gauge coupling matrix . Thus, even if at one scale is diagonal, in generaland non-zero off-diagonal Uentries are generated by renormalization effects. One shows [187] that in the case the abelian gauge couplings are at a given scale diagonal equal (i.e. there is a (1) unification), there may exist a (scale independent) gauge field basis such2 that the abelian interactions remain to all orders diagonal along the RGE trajectory . dgrb In general, the renormalization of theg abelianraβab ; part of the gauge interactions is determined by dt

d = (2.9) β Z / : where, as a consequence of gaugeab invariance,dt ab 1 2 Z 3 = log (2.10) 3 r with denoting the gauge-boson wave-functiongkb Qkgrb ; renormalization matrix. In order to further simplify the notation it is convenient≡ to introduce the “reduced” couplings [187]

dgkb (2.11) gkaβab : that evolve according to dt k U =β U (2.12) The index labels the ﬁelds (fermions and scalars) that carry (1) charges. In terms of the reduced couplings the -function that governs the (1) running up β κ g g η g g to twoab loops isπ given byfa [179,fb 180, 181]sa sb 1 2 4 1 = κ gfagfbg + gfagfbg C Fq η gsagsbg gsagsbg C Sq (2.13) (4π) 3 fc 3 q sc q h 2 2 2 2 i 4κ 2 2 2 + gfagfb Y Y+† ; ( ) + + ( ) − (4π) 2 2 Tr 1 (4 ) Alternatively one may work with oﬀ-diagonalβ kinetic terms while keeping the gauge interactions diagonal2 [186]. Vanishing of the commutator of the -functions and their derivatives is needed [188]. Chapter 2. Intermediate scales in nonsupersymmetric SO unification

78 f (10) s U c C Rp where repeated indices are summed over, labelling fermionsGq( ), scalars ( ) and (1) 2 gaugeU groups ( ). The terms proportional to the quadratic Casimir ( ) represent the two-loop contributions of the non abelian components of the gauge group to the (1) gauge coupling renormalization. Correspondingly,U using the notation of Eq. (2.11), an additional two-loop term that represents the renormalization of the non abelian gauge couplings induced at two loops g by the (1) gauge ﬁelds is to bep added to Eq. (2.2), namely βp κ gfcS Fp η gscS Sp : π2 2 2 h 2 2 i ∆ = 4 4 ( f ) + F (s ) S (2.14)κ (4 ) p p ; η ≡; ≡ In1 Eqs. (2.13)–(2.14), we use the abbreviation 1 and and, as before, = 1 2 for Dirac and Weyl fermions, while = 1 2 for complex and real scalar ﬁelds 2.2.3respectively. Some notation U

When at most one (1) factor is present, and neglecting the Yukawa contributions, the dαi− ai bij two-loop RGEs can be conveniently written as αj ; dt 1 − π − π

2 αi g / π β = ai bij SO (2.15) i 2 8 2 where = 4 . The -coefficients αandj for the relevant (10) chains are given in Appendix A. Substituting the one-loop solution forai intob theij right-hand side of Eq. (2.15) one αi− t αi− t ωjt ; obtains − − π π − 1 1 ˜ ωj aj αj / π ( ) bij (0)bij =/aj + log (1 ) (2.16) ωj t < 2 4 β bwhereij = (0) (2 ) and ˜ = . The analytic solution in (2.16) holds at two loops (for 1) up to higher orderβ effects. A sample of the rescaled -coefficients ˜ is given, for the purpose of comparison with previous results, in Appendix A.

We shall conveniently writeβab the -functiongsa γ insr Eq.grb ; (2.13), that governs the abelian mixing, as π 1 γsr = 2 (2.17) (4 U) where include both one- and two-loop contributions. Analogously, the non-abelian gp beta function in Eq. (2.2), includingβ thep (1) contributionγp : in Eq. (2.14), is conveniently written as π2 = 2 (2.18) (4 ) 2.2. Two-loop gauge renormalization group equations

79 γp SO

The functions for the (10) breaking chains considered in this work are reported in Appendix A.1. Finally, the Yukawa term inY Eq.Fp (2.5),ypk andY correspondinglykYk† ; in Eq. (2.13), can be written as Yk 4 k ×( ) = Tr k (2.19) where are the “standard” 3 3 Yukawa matricesSO in the family space labelledy bypk the flavour index . The trace is taken over family indices and is summed over the different Yukawa terms present at each stage of (10) breaking. The coefficients 2.2.4are given One-loop explicitly in matching Appendix A.2

The matchingG conditions between effective theoriesGp in the framework of dimensional regularization have been derived in [189, 190]. Let us consider first a simple gauge group spontaneously broken into subgroups . Neglecting terms involving logarithms of mass ratios which are assumed to be subleading (massive states clustered C Gp C G near the threshold), the one-loopαp− matching forαG− the gauge: couplings can be written as − π − π 1 2 1 2 ( ) ( ) = Gp (2.20) U X 12 U12Y U Let us turnTY to the case when several non-abelian simple groups (and at most one (1) ) spontaneously break whilst preserving aTX (1) charge. The conserved (1) generator can be written in terms of the relevant generators of the various Cartan TY piTi ; subalgebras (and of the consistently normalized ) as p i p X i = (2.21) 2 P C G where =3 1, and runs over the relevant (and )i indices. The matching condition αY− pi αi− ; is then give by − π 1 i 2 1 2 X ( ) = (2.28) 12 φ TY φ 3 h i h i This is easily proven at tree level [191]. Let us imagine that the gauge symmetry is spontaneously broken by the VEV of an arbitrary setDµ ofφ scalar∂µφ fieldsigiTi Aµ,i suchφ ; that = 0. Starting from the covariant derivative = + (2.22) µ gigj φ † TiTj φ ; we derive the gauge boson mass matrix ij h i h i 2 AY q A = µ i µ i (2.23) which has a zero eigenvector correspondingµij qj to the masslessq gaugej :field = , where j 2 X 2 = 0 with = 1 (2.24) Chapter 2. Intermediate scales in nonsupersymmetric SO unification

80 i X C (10) N U M 2 where forU = , if present,M

U ::: U N G ::: GN′ ⊗ ⊗ ⊗ ⊗ ⊗ T where =1U :::andU =1M M. N N′ gg − ⊗ ⊗ ≤ 1 1 The general case of a gauge groupN N(1)′ N N′(1) spontaneously1 1 × broken to (1) (1) with + is taken care of by replacing ( ) in T T C Gp Eq. (2.30) with the block-diagonalGG − ( + gg) (− ;+ g − ) matrix p − π 1 1 2 2 ( ) ( ) = Diag ( ) 2 (2.32) 48 thus providing, together with the extendedq Eq. (2.29) and Eq. (2.30), a generalization of Eq. (2.28). It’s easy to see then that the components of are − qi Npi/gi N pi/gi 1 ; ≡ 2 i ! X 2 Y = with ( ) (2.25) Aµ

g A ψγµT ψ g q AY ψγµT ψ : : : Np AY ψγµT ψ : : : NAY ψγµT ψ ::: ; and from thei couplingµ i ofi toi i fermionsµ i i µ i µ Y = + = + = + (2.26) − we conclude that gY N pi/gi 1 : ≡ 2 i ! X 2 = ( ) (2.27) 2.3. Numerical results

2.3 Numerical results 81

At one-loop, and in absence of theU (1) mixing, the gauge RGEs are not coupled and the unification constraints can be studied analytically. When two-loop effects are included (or at one-loop more than one (1) factor is present) there is noU closed solution and one must solve the system of coupled equations, matching all stages between the weak and unification scales, numerically. On the other hand (when no (1) mixing is there) one may take advantage of the analytic formula in Eq. (2.16). The latter turns out to provide,− for the cases here studied, a very good approximation to the numerical solution.3 The discrepancies with the numerical integrationM doU notM generallyM exceed the 10 level. αU 2 1 We performM aZ scan over the relevant breaking scales , and and the value of the grand unified coupling and impose the matching with the SM gauge couplings at the scale requiring a precision at the per mil level. This is achieved α αi by minimizing the parameter δ i − ; v 3 thαi 2 u i uX = t (2.33) αi =1 MZ αi th where denote the experimental values at and are the renormalized couplings MobtainedZ : from unification. The input values for the (consistentlyα : normalized): gauge; SM couplings at the scale ± = 91 19 GeV are [80] α : : ; 1 ± α = 0:016946 : 0 000006; 2 ± = 0 033812 0 000021 (2.34) 3 = 0 1176 0 0020 αem− : : ; corresponding to the electroweak scale parameters± θW1 : : : ± 2 = 127 925 0 016 MZ sin = 0 23119 0 00014 (2.35) All theseα ; data refer to the modified minimally subtracted (MS) quantities at the scale. σ α 1 2 For we shall consider only the central values while weM resortM to scanningMU over 3 the whole 3 domain for when a stable solution is not found. n 1 2 TheM results,/ n i.e. the positionsM / ofnU the intermediateMU / scales , andnU n shall be 1 reported in termsn of decadic logarithms of their values in units of GeV, i.e. = 10 1 2 10 2 10 2 log ( GeV), = log ( GeV), = log ( GeV). In particular, , are given 1 as functions of for eachαU breaking pattern and for different approximations in the loop expansion. Each of the breaking patterns is further supplemented by the relevant range of the values of . Chapter 2. Intermediate scales in nonsupersymmetric SO unification

2.3.182 R B L mixing (10) U ⊗U − XII U U (1)R U (1)B L − The chains VIII to require consideration of the mixing between the two (1) factors. While (1) and (1) do emerge with canonicalscale independent diagonal kinetic terms, being the remnants of the breaking of non-abelian groups, the corresponding gauge couplings are at the onset different in size. In general, no orthogonal rotations of charges and gauge fields exist that diagonalize the gauge interactions to all orders along the RGE trajectories.M According to the discussion in Sect. 2.2, off-diagonal gauge R;B L couplingsQ arise− at the one-loop level that must be accounted for in order to perform 1 the matching at the scale with the standard hypercharge. The preserved direction Y R B L in the charge space is givenQ by Q Q − ; r r 3 2 = + (2.36) 5 5 R B L B L Q T R Q − − : where r 3 3 = and = (2.37) 2 2 T − T The matching of the gauge couplingsgY− isP thengg obtainedP ; from Eq. (2.30) 1 2 = (2.38) P ; with r r ! 3 2 = (2.39) gR;R 5 gR;B5 L g − : and gB L;R gB L;B L ! − − − = (2.40) U When neglecting the off-diagonal terms, Eq. (2.38) reproduces the matching condition used in Refs. [174, 100, 175, 64]. For all other cases, in which only one (1) factor 2.3.2is present, Two-loop the matching results relations (purely can be gauge) read off directly from Eq. (2.20) and Eq. (2.28).

nU n n The results of the numerical analysisH areH organized as follows: Fig. 2.1 and Fig. 2.2 2 1 show the values of andH asH functions of for the pure gauge running (i.e. no Yukawa interactions), in the 126 and 16 case respectively. The differences between Uthe patterns for the 126 and 16 setups depend on the substantially different scalar content. The shape and size of the various contributions (one-loop, with and without (1) mixing, and two-loops) are compared in each figure. The dissection of the RGE 2.3. Numerical results

results shown in the figures allows us to compareαU− our results with those of Refs.n [174, 100, 175, 64]. φ 1 1 Table 2.3 shows the two-loop126 values of in the allowed region for . The contributions of the additional multiplets, and the Yukawa terms are discussed separately in Sect. 2.3.3 and Sect. 2.3.4, respectively. With the exception of a few singular cases detailed therein, these effects turn out to be generally subdominant. As already mentioned in the introduction, two-loop precision in a GUT scenario makes sense once (one-loop) thresholds effects are coherently taken into account, as their effect may become comparable if not larger than the two loop itself (the argument becomes stronger as the number of intermediate scales increases). On the other hand, there is no control on the spectrum unlessSO a specific model is studied in details. The purpose of this chapter is to set the stage for such a study by reassessing and updating the general constraints andβ patterns that (10) grand unification enforces on the spread of intermediate scales. bij The one and two-loop -coefficients used in the present study are reported in Ap- pendix A. Table A.5 in the appendix shows the reduced e coefficients for those cases where we are at variance with Ref. [100]. U U R ⊗ U B L SO One− of the largest effects in the comparison with Refs. [174, 100, 175, 64] emerges at one-loopXII, and it is due to the implementation of the (1) gauge mixing4 n n whenU (1)αU (1) appearsn as an intermediate stage of the (10) breaking . This affects chains 2 VIII to and it exhibits itself in the exact (one-loop) flatness of , and as 1 C L R B L functions of . − The rationaleβ for such a behaviour is quite simple. When considering the gauge C L R B L C L Y coupling renormalization in the− 3 2 1 1 stage, no effect at one-loop appears in the non-abelian -functions due to the abelian gauge fields.β On the other hand, the Higgs fields surviving at the 3 2 1 1 nstage,nU responsibleαU forn the breakingn to 3 2 1 ,n are< (by n construction) SM singlets. Since the SM one-loop U-functionsR U areB L not affected by their 2 1⊗ 2 − 1 2 presence, the solution found forU Y , and in the = casen holds for as well. Only by performing correctly the mixed (1) (1) gauge running and 1 the consistent matching with (1) one recovers the expectedU flatness of the GUT solution. U R In this respect, it is interesting to notice that the absence of (1) mixing in Refs. [174, 100, 175, 64] makes the argument for the actual possibility of a light (observable) (1) gauge boson an “approximate" statement (based on theSU approxL imateSU flatnessC β of the solution). One expects this feature to break atU two-loops.R U TheB L (2) and (3) -functions ⊗ − are affected at two-loops directly by the abelian gauge bosons via Eq. (2.14) (the Higgs multiplets that are responsible for the (1) (1) breaking do not enter through the Yukawa interactions). The net effect on the non-abelian gauge running is related 4 The lack of abelian gauge mixing in Ref. [64] was first observed in Ref. [192]. Chapter 2. Intermediate scales in nonsupersymmetric SO unification

n n 84 2 U n2 nU n2 nU (10) 16.0 17.0 16.5 16 15.5 16.0 14 15.0 15.5

14.5 15.0 12 14.5 14.0 10 14.0 n1 13.5 n1 13.5 n1 8.5 9.0 9.5 10.0 10.5 11.0 11.5 10 11 12 13 14 9 10 11 12 13 14

n n (a) Chain Ia n n (b) Chain IIa (c) Chain IIIa 2 U 2 U n2 nU 15.5 16 16 15 15.0

14 14 14.5 13 12 12 14.0

10 11 n1 n1 13.5 n1 8.5 9.0 9.5 10.0 10.5 11.0 10.6 10.8 11.0 11.2 11.4 11.6 11.8 12.0 11.0 11.5 12.0 12.5 13.0 13.5 14.0

(d) Chain IVa (e) Chain Va (f) Chain VIa n2 nU n2 nU n2 nU 16.5 16 16 16.0 15 15.5 14 14 15.0 13 12 14.5 12 14.0 10 11 13.5 10 8 13.0 n1 n1 9 n1 11.0 11.5 12.0 12.5 13.0 13.5 14.0 3 4 5 6 7 8 9 4 6 8 10 IX

(g) Chain VIIa (h) Chain VIIIa (i) Chain a n2 nU n2 nU 15.5 15

15.0 14

14.5 13

14.0 12 11 13.5 10 13.0 n1 n1 4 6 8 10 12 14 4 6 8 10

XIa (k) Chain XIIa (j) Chain

nU n The values of (red/upper branches) and (blue/lower branches) are shown as functions n H n n of for the pure gauge running in the case. The2 bold black line bounds the region ≤ . Figure1 2.1: 1 2 From chains Ia to VIIa the short-dashed lines126 represent the result of one-loop running while the solid ones correspond to the two-loop solutions. For chains VIIIa to XIIa the short-dashed lines represent U R U B L the one-loop results without the ⊗ − mixing, the long-dashed lines account for the complete one-loop results, while the solid(1) lines represent(1) the two-loop solutions. The scalar content at eachφ stage corresponds to that considered in Ref. [64], namely to that reported in Table 2.2 without the SO 126 multiplets. For chains I to VII the two-step breaking consistent with minimal ﬁne tuning is n nU → recovered in the limit. No solution is found(10) for chain Xa. 2 2.3. Numerical results

n2 nU n2 nU n2 nU 17 17 17

16 16 16

15 15 15

14 14 14

13 n1 13 n1 13 n1 10 11 12 13 14 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 10 11 12 13 14

(a) Chain Ib (b) Chain IIb (c) Chain IIIb

n n 2 U n2 nU n2 nU 17 16.0 16.0

16 15.5 15.5 15 15.0 15.0 14 14.5 14.5 13 14.0 14.0 12 11 13.5 13.5 10 n1 13.0 n1 13.0 n1 10.0 10.2 10.4 10.6 10.8 11.0 11.2 11.4 12.0 12.5 13.0 13.5 14.0 12.0 12.5 13.0 13.5 14.0

(d) Chain IVb (e) Chain Vb (f) Chain VIb

n2 nU n2 nU n2 nU 16.0 17 16 16 15.5 15 15.0 14 14 14.5 13 12 14.0 12

13.5 10 11 13.0 n1 n1 10 n1 13.50 13.55 13.60 13.65 13.70 13.75 13.80 4 6 8 10 4 6 8 10 12

(g) Chain VIIb (h) Chain VIIIb (i) Chain IXb

n2 nU n2 nU n2 nU 16 15.5 15 15 15.0 14 14.5 14 14.0 13 13 13.5 12 12 n1 13.0 n1 n1 4 6 8 10 12 4 6 8 10 12 14 4 6 8 10 12

(j) Chain Xb (k) Chain XIb (l) Chain XIIb

H Same as in Fig. 2.1 for the case. Figure 2.2: 16 Chapter 2. Intermediate scales in nonsupersymmetric SO unification

86 (10) αU− αU− : ; 1 : : ; 1 : Chain : ; : Chain : ; : Ia [45:5; 46:4] Ib [45:7; 44:8] IIa [43:7; 40:8] IIb [45:3; 44:5] IIIa [45:5; 40:8] IIIb [45:7; 44:5] IVa [45:5; 43:4] IVb [45:7; 45:1] Va [45:4; 44:1] Vb [44:3; 44:8] VIa [44 1 :41 0] VIb [44 3 :44 2] VIIa [45 4 :41 1] VIIb [44 8 :44 4] IXa IXb VIIIa 45 4 VIIIb 45:6 Xa Xb 42:8 44:3 XIa XIb : 44:8 XIIa XIIb 38 7 41 5 αU− n αU− n Two-loop values of in44 the1 allowed region for 44. From3 chains I to VII, is Table 2.3: 1 1 dependent and its range is given in square brackets for the minimum1 (left) and the maximum (right)1 n αU− n value of respectively. For chains VIII to XII, depends very weekly on (see the discussion on U 1 mixing1 in the text). No solution is found for chain Xa. 1 (1) U R U B L to the difference between the contribution of the and − gauge bosons and that of the standard hypercharge. We checked that such a difference is always a small (1) (1) SU L SU C βfraction (below 10%) of the typical two-loopn contributions to the and -functions. As a consequence, the flatness of the GUT solution is at a very high − (2) (3) accuracy ( ) preserved at two-loops1 as well, as the inspection of the relevant chains 3 in Figs. 2.1–2.2 shows. 10 n Still at one-loop we find a sharp disagreement inn the< :range of chain XIIa, with respect to the result of Ref. [64]. The authorsn < find: 1 , while strictly following their procedure and assumptions we find 1(the updated one- and two-loop 5 3 results are given in Fig. 2.1k). As we shall see,1 this difference brings chain XIIa back 10 2 among the potentially realistic ones. As far as two-loop effects are at stakes, their relevance is generally related to the length of the running involving the largest non-abelian groups. On the other hand, n nU n there are chains where and have a strong dependence on (we will refer to them as to “unstable" chains)2 and where two-loop corrections affect1 substantially the one-loop results. Evident examples of such unstable chains are Ia, IVa, Va, IVb, and n nU VIIb. In particular, in chain Va the two-loop effectsn flip the slopes of and , that implies a sharp change in the allowed region for . It is clear that when2 dealing with these breaking chains any statement about their viability1 should account for the details of the thresholds in the given model. C L R B L − In chains VIII to XII (where the second intermediate stage is , two-loopnU effects are mild and exhibit the common behaviour of lowering the GUT scale ( ) 3 2 1 1 2.3. Numerical results 87

n while raising (with the exception of Xb and XIa,b) the largest intermediate scale ( ). The mildness ofn two-loop corrections (no more that one would a-priori expect) is strictly2 related to the ( ) flatness of the GUT solution discussed before. n nU n n Worth mentioning1 are the limitsG ∼ and ∼ . While the former is equivalent to neglecting theSO first stageG andSM2 to reducing1 effectively2 the three breaking steps to just two (namely Ï Ï ) with a minimal fine tuning in the scalar sector, 2 care must be taken of the latter. One may naively expect that the chains with the same (10)n n 1 SO G SM G2 should exhibit for ∼ the same numerical behavior ( Ï Ï ), thus clustering the chains1 (I,V,X),2 (II,III,VI,VII,XI) and (IV,IX). On the other hand, one (10) 2 must recall that the existence of G1 and its breaking remain encoded in the G2 stage through the Higgs scalars that are responsible for the G2→G1 breaking.n n This is why ∼ the chains with the same G2 are not in general equivalentn n inU the limit. The numerical features of the degenerate patterns (with ∼ ) can1 be crosschecked2 among the different chains by direct inspection of Figs.2 2.1–2.2 and Table 2.3. In any discussion of viability of the various scenarios the main attention is paid to the constraints emerging from the proton decay. In nonsupersymmetric GUTs this process is dominated by baryond number violating gauge interactionsB L , inducing at low energiesSO a set of effective operators that conserve − (cf. Sect. 1.6.1). In the scenarios we consider here such gauge bosons are integrated out at the unification = 6 nU (10) scale and therefore proton decay constrains from below. Considering the naive estimate of the inverse lifetime of the proton in Eq. (1.234), the present experimental τ p π e > : nU & : αU− limit → × yr [11] yields , for . Taking the 0 + 33 1 results in Figs. 2.1–2.2 and Table 2.3 at face value the chains VIab, XIab, XIIab, Vb and ( ) 8 2 10 15 4 = 40 VIIb should be excluded from realistic considerations. On the other hand, one must recall that once a specific model is scrutinized in detail there can be large threshold corrections in the matching [176, 177, 178], that can easily move the unification scale by a few orders of magnitude (in both directions). In particular, as a consequence of the spontaneous breaking of accidental would-be global symmetries of the scalar potential, pseudo-Goldstone modes (with masses further suppressed with respect to the expected threshold range) may appear in the scalar spectrum, leading to potentially large RGE effects [47]. Therefore, we shall follow a con- servative approach in interpreting the limits on the intermediate scales coming from a simple threshold clustering. These limits, albeit useful for a preliminary survey, may not be sharply used to excludeB L marginal but otherwise well motivated scenarios. Below the scale of the − breaking, processesB that violate separately the barion or the lepton numbers emerge. In particular, effective interactions give rise to the phenomenon of neutron oscillations (for a recent review see Ref. [104]). Present τN > ∆ =2 bounds on nuclear instability give years, which translates into a bound τn n > 32 on the neutron oscillation time − sec. Analogous limits come from direct 108 B reactor oscillations experiments. This¯ sets a lower bound on the scale of d 10 nonsupersymmetric ( ) operators that varies from 10 to 300 TeV depending on ∆ = 2 = 9 Chapter 2. Intermediate scales in nonsupersymmetric SO unification 88 (10) model couplings. Thus, neutron-antineutron oscillations probeB scalesd far below the unification scale. In a supersymmetricB L context the presence of operators softens the dependence on the − scale and for the present bounds the typical limit ∆ =2 =7 goes up to about GeV. 7 L Far more reaching in scale sensitivity are the neutrino masses emerging 10 B L R δR from the see-saw mechanism. At the − breaking scale the ( ) scalars acquire L L ∆ = 2 126 16 ( ) VEVs that give a Majorana mass to the right-handedT T neutrinos. Once d ℓ H∆ C H ℓ / L the latter are integrated out, operators of the form generate ∆ =2 ∆ =1 light Majorana neutrino states in the low energy theory. 2 2 T =5 mν ( ) ( ) ΛYDMR− YD v In the type-I seesaw, the neutrino mass matrix is proportional to 1 2 where the largestMR entryM in the Yukawa couplings is typically of the order of the top quark one and ∼ . Given a neutrino mass above the limit obtained from atmo-< spheric neutrino oscillations1 and below the eV, one infers a (loose) range GeV M < 13 GeV. It is interesting to note that the lower bound pairs with the cosmological 15 10 limit1 on the D-parity breaking scale (see Sect. 2.1). 10 In the scalar-triplet induced (type-II) seesawSU theL evidence ofH the neutrino mass entails a lower bound on the VEV of the heavy triplet in . This translates into an upper bound on the mass of the triplet that depends on the structure of the relevant (2) 126 Yukawa coupling. If bothM type-I as well as type-II contribute to the light neutrino mass, the lower bound on the scale may then be weakened by the interplay between the two contributions. Once1 again this can be quantitatively assessed only when the vacuum of the model is fully investigated. B L H − Finally, it is worth noting that if the breakdown is driven by , them ele-ν mentary triplets couple to the MajoranaG currentsSM at the renoMrmalizable level and is → 126 directlyn sensitive tom theν position of the threshold . On the other hand, the -dependence of is loosened in the b-type of chains due to1 the non-renormalizable 1 F F H H / 1 nature>MU of the relevant effective operator , where the effective scale accounts for an extra suppression. 16 16 16 16 Λ With these considerations at hand, the constraints from proton decay and the see- Λ C L RP SOsaw neutrino scale favor the chains II, III and VII, which all share in the first breaking stage [141]. On the other hand, our results rescue from oblivion other 4 2 2 potentially interesting scenarios that, as we shall expandB L upon shortly, are worth of in (10) − depth consideration. In all cases, the bounds on the U scaleR enforced by the see-saw neutrino mass excludes the possibility of observable gauge bosons. 2.3.3 The Higgs multiplets (1) φ 126

As mentioned in Sect. 2.1.1, in order to ensure a rich enoughSU YukawaL sector in realistic models there may be the need to keep more than one Higgs doublet at intermediate scales, albeit at the price of an extra ﬁne-tuning. A typical example is (2) the case of a relatively low Pati-Salam breaking scale where one needs at least a pair 2.3. Numerical results 89

n2 nU n2 nU n2 nU 17 18 16 16 16 14 14 15 12 12 14 10 10 8 n1 n1 13 n1 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 10.0 10.2 10.4 10.6 10.8 11.0 11.2 11.4 10 11 12 13 14

(a) Chain Ia (b) Chain Va (c) Chain VIIa

φ Example of chains with sizeable effects (long-dashed curves) on the position of the Figure 2.3: 126 intermediate scales. The solid curves represent the two-loop results in Fig. 2.1. The most dramatic effects appear inβ the chain Ia, while moderate scale shifts affect chain Va (both “unstable" under small variations ofφ the -functions). Chain VIIa, due to the presence of two PS stages, is the only "stable” chain with visible effects. 126

SU L SU R SU C of ⊗ bidoublets with different quantum numbers to transfer the information about the PS breakdown into the matter sector. Such additional Higgs (2) (2) φ (4) multiplets are those labelled by in Table 2.2. 126 φ SU C Table 2.4 shows the effects of including at the stages of the relevant 126 breaking chains. The two-loop results at the extreme values of the intermediate scales, φ (4) with and without the multiplet, are compared. In the latter case the complete 126 functional dependence among the scales is given in Fig. 2.1. Degenerate patterns with only one effective intermediate stage are easily crosschecked among the different chains in Table 2.4. In most of the; cases,; the numerical; ; results do not exhibit a sizeable dependence on the additional (or ) scalar multiplets. The reason can be read 1 off Table A.6 in Appendix126 A and it rests2 126 on an accidental approximate coincidence of φ (15 2 2) SU(15 2C+ ) SU L;R the contributions to the and one-loop beta coefficients (the same 126 C L R argument applies to the case). (4)C L R (2) a T Considering for instance the stage, one obtains × × , a T 4 2 1 1 αU 16 and × × , that only slightly affects the value4 of3 (when2 the PS3 1 4 2 2 ∆ = 4 (15) = scale is2 low3 enough), but2 has generally a negligible effect on the intermediate scales. ∆ = 30 (2) = 5 n ;UAnn exception to this argument is observed in chainsβ Ia and Va that, due to their slopes, are most sensitive to variations of the -coefficients. In particular, the φ n nU inclusion2 1 of in the Ia chain flips at two-loops the slopes of and so that the ( n) nU 126 n limit (i.e. no G2 stage)n is obtained forφ the maximal value of2 (while the same happens2 for the minimum if there is no ). 1 = 126 φ Fig. 2.3 shows three template1 cases where the effects are visible. The highly 126 unstableφ Chain Ia shows, as noticed earlier, the largest effects. In chain Va the effects of are moderate. Chain VII is the only "stable" chain that exhibits visible effects on 126 Chapter 2. Intermediate scales in nonsupersymmetric SO unification 90 (10)

n n nU αU− Chain 1 Ia [9.50,1 10.0] [16.2,2 10.0] [16.2, 17.0] [45.5, 46.4] [8.00, 9.50] [10.4, 16.2] [18.0, 16.2] [30.6, 45.5] IIa [10.5, 13.7] [15.4, 13.7] [15.4, 15.1] [43.7, 40.8] [10.5, 13.7] [15.4, 13.7] [15.4, 15.1] [43.7, 37.6] IIIa [9.50, 13.7] [16.2, 13.7] [16.2, 15.1] [45.5, 40.8] [9.50, 13.7] [16.2, 13.7] [16.2, 15.1] [45.5, 37.6] Va [11.0, 11.4] [11.0, 14.4] [15.9, 14.4] [45.4, 44.1] [10.1, 11.2] [10.1, 14.5] [16.5, 14.5] [32.5, 40.8] VIa [11.4, 13.7] [14.4, 13.7] [14.4, 14.9] [44.1, 41.0] [11.2, 13.7] [14.5, 13.7] [14.5, 14.9] [40.8, 38.1] VIIa [11.3, 13.7] [15.9, 13.7] [15.9, 14.9] [45.4, 41.1] [10.5, 13.7] [16.5, 13.7] [16.5, 15.0] [33.3, 38.1] XIa [3.00, 13.7] [13.7, 13.7] [14.8, 14.8] [38.7, 38.7] [3.00, 13.7] [13.7, 13.7] [14.8, 14.8] [36.0, 36.0] XIIa [3.00, 10.8] [10.8, 10.8] [14.6, 14.6] [44.1, 44.1] [3.00, 10.5] [10.5, 10.5] [14.7, 14.7] [39.8, 39.8] φ Impact of the additional multiplet (second line of each chain) on those chains that Table 2.4: C L R C L R 126 contain the gauge groups or as intermediate stages, and whose breaking to the SM is H n nU αU− obtained via a representation. The values of , and are showed for the minimum and 4 2 n2 4 2 1 1 2 maximum values126 allowed for by the two-loop analysis. Generally the eﬀects on the intermediate 1 scales are belowβ the percent level, with the exception of chains Ia and Va that are most sensitive to variations of the -functions. 2.3. Numerical results 91 the intermediate scales. This is due to the presence of two full-ﬂedged PS stages. 2.3.4 Yukawa terms

The eﬀects of the Yukawa couplings can be at leading order approximated by constant ai ai ai′ ai ai negative shifts of the one-loop coeﬃcients → with

ai yik Yk Y=k† : + ∆ − π Tr (2.41) 1 2 ai ∆ = The impact of on the position of the(4 unification) scale and the value of the unified coupling can be simply estimated by considering the running induced by the Yukawa ∆ t t couplings from a scale up to the unification point ( ). The one-loop result for the αi− t αj− t change of the intersection of the curves corresponding to and reads (at ai =0 1 1 the leading order in ): ( ) ( ) ai aj ∆ tU π − αj− t αi− t ::: ai aj − (2.42) − 1 1 ∆ ∆ ∆ =2 2 ( ) ( ) + and ( ) ai aj ai aj ai aj αU− − αj− t αi− t ::: ai aj − ai aj − (2.43) 1 − − 1 1 1 ∆ + ∆ ( + )(∆ 2 ∆ ) ∆i j= ( ) ( )ai+ 6 2 ( ) for any . For simplicity we have neglected the changes in the coefficientsai dueaj to crossing intermediate thresholds. It is clear that for a common change = αU− the unification scale is not affected, while a net effect remains on . In all cases, αj− t αi− t 1 ∆ = ∆ the leading contribution is always proportional to − (this holds exactly for tU 1 1 ). ( ) ( ) In order to assess quantitatively such effects we shall consider first the SM stage ∆ thatn accounts for a large part of the running in all realistic chains. The case of a low scale leads, as we explain in the following, to comparably smaller effects. The impact1 of the Yukawa interactions on the gauge RGEs is readily estimated assuming YU YU† only the up-type Yukawa contribution to be sizeable and constant, namely Tr ∼ . ai − yiU yiU This yields ∼ − × , where the values of the coefficients are given in i j 3 a : − a : −1 Table A.7. For SMand oneSM obtains ∼ − × and ∼ − × ∆ a6 10 a 2 2 respectively. Since and − , the1 first term in (2.43) dominates2 and one αU− : =1 41=2 αU−19∆ 1 1 10 ∆ αU− /αU−0 9 10: finds ∼ . For1 a typical10 value2 of 6 ∼ this translates into ∼ . 1 tU = = n1U − 1 1 The impact on is indeed tiny, namely ∼ − × . In both cases the estimated ∆ 0 04 40 2 ∆ 0 1% effect agrees to high accuracy with the actual numerical behavior we observe. ∆ 1 10 The effects of the Yukawa interactions emerging at intermediate scales (or of a non- YD YD† β negligible Tr in a two Higgs doublet settings with large ) can be analogously SO YN Y † YU Y † N ∼ U accounted for. As a matter of fact, in the type of modelstan Tr Tr (10) Chapter 2. Intermediate scales in nonsupersymmetric SO unification 92 (10) YU YN due to the common origin of and . The unified structure of the Yukawa sector ai k yik yields therefore homogeneous factors (see the equality of in Table A.7). This provides the observed large suppression of the Yukawa effectsP on threshold scales and ∆ unification compared to typical two-loop gauge contributions. In summary, the two-loop RGE effects due to Yukawa couplings on the magnitude of the unification scale (and intermediate thresholds) and the value of the GUT gauge coupling turnn outU to beαU very small. Typically we observe negative shifts at the per-mil level in both and , with no relevant impact on the gauge-mediated proton decay rate. 2.3.5 The privilege of being minimal

With all the information at hand we can finally approach an assessment of the viability of the various scenarios. As we have argued at length, we cannot discard a marginal unification setup without a detailed information on the fine threshold structure. ObtainingSO this piece of information involve the study of the vacuum of the model, and for GUTs this is in general a most challenging task. In this respect supersymmetry helps: the superpotential is holomorphic and the couplings in the renor- (10) malizable caseF are limitedD to at most cubic terms; the physical vacuum is constrained by GUT-scale - and -flatness and supersymmetry may be exploited to studying the fermionic rather than the scalar spectra. SO It is not surprising that for nonsupersymmetric , only a few detailed studies of the Higgs potential and the related threshold effects (see for instance Refs. [56, 193, (10) 58, 59, 194]) are available. In view of all this and of the intrinsic predictivity related to minimality, the relevance of carefully scrutinizing the simplest scenarios is hardly overstressed. SO H The most economical Higgs sectorH includesH the adjoint , that provides the breaking of the GUT symmetry, either or , responsible for the subsequent B L H (10) 45 − breaking, and , participating to the electroweak symmetry breaking. The H 16H 126 latter is needed together with or in order to obtain realistic patterns for 10 the fermionic masses and mixing. Due to the properties of the adjoint representation 16 126 this scenario exhibits a minimal number of parametersSO in the Higgs potential. In the current notation such a minimal nonsupersymmetric GUT corresponds to the chains VIII and XII. (10) From this point of view, it is quite intriguing that our analysis of the gauge unification constraints improves the standing of these chains (for XIIa dramatically) withH respect to existing studies. In particular, considering the renormalizable setups ( ), we find n : nU : αU− : n : for chain VIIIa, ≤ , and (to be compared to ≤ given 1 126 in Ref. [64]). This1 is due to the combination of the updated weak scale data1 and two loop 9 1 = 16 2 n = 45: 4nU : αU− : 7 7 running effects. For chain XIIa we find ≤ , and , showing n : 1 a dramatic (and pathological) change from1 ≤ obtained in [64]. Our result sets 10 8 = 14 6 = 44 1 1 5 3 2.3. Numerical results 93

B L − the scale nearby the needed scale for realistic light neutrino masses. H We observe non-negligible two-loop effects for the chains VIIIb and XIIb ( ) as n : nU : αU− : B L well. For chain VIIIb we obtain ≤ , and (that lifts the − nU 1 ?? 16 scale while preserving well above1 the proton decay bound Eq. ( )). A similar shift n 10 5 = 16n 2 : nU= 45 6 : αU− : in is observed in chain XIIb where we find ≤ , and . nU : 1 As we1 have already stressed one should not too1 readily discard as being 12 5 = 14 8 = 44 3 incompatible with the proton decay bound. We have verified that reasonable GUT nU = 14 8 threshold patterns exist that easily lift above the experimental bound. For all these chains D-parity is broken at the GUT scale thus avoiding any cosmological issues (see the discussion in Sect. 2.1). n n SO As remarked in Sect. 2.3.2, the limit leads to an effective two-step → G2 → SM breaking with a non-minimal1 set of2 surviving scalars at the G2 stage. As = (10) a consequence, the unification setup for the minimal scenarino cann beU recovered (with the needed minimal fine tuning) by considering the limit in those chains C L R B L C L R among I to VII where G1 is either − or (see2 Table 2.1). From inspec- = SO tion of Figs. 2.1–2.2 and of Table 2.3, one reads the following results: for ÊÏ C L R B L 3 2 2 1 4 2 1 − → SM we find (10) 45 n : nU : αU− : a 3 2•2 1 , and (case ), 1 n1 : nU : αU− : b • =9 5 = 16 2 and = 45 5 (case ), 1 1 SO C L R while for= 10 8 ÊÏ= 16 2 → SM= 45 7

n (10): nU45 4 :2 1 αU− : a • , and (case ), 1 n1 : nU : αU− : b • = 11 4, = 14 4 and = 44 1 (case ). 1 1 We observe= 12 that6 the= 14 patterns6 are= quite 44 3 similarn n to those of the non-minimalφ setups obtained from chains VIII and XII in the limit. Adding the multiplet, as a 126 required by a realistic matter spectrum in case1 , does2 not modify the scalar content in C L R B L H H − = the case: only one linear combination of the and bidoubletsC L R (see Table 2.2) is allowed by minimal fine tuning. On the other hand, in the case, 3 2 2 1 10 126 αU− : the only sizeable effect is a shift on the unified coupling constant, namely 4 2 11 (see the discussion in Sect. 2.3.3). B L= 40 7 In summary, in view of realistic thresholds effects at the GUT (and − ) scale and of a modest fine tuning in the see-saw neutrino mass, we consider both scenarios worth of a detailed investigation. Chapter 2. Intermediate scales in nonsupersymmetric SO unification 94 (10) Chapter 3

The quantum vacuum of the minimal SO GUT

3.1(10) The minimal SO(10) Higgs sector SO In this chapter we consider a nonsupersymmetric setup featuring the minimal Higgs content sufficient to trigger the spontaneous breakdown of the GUT symmetry (10) down to theH standardH electroweak model. Minimally,H the scalar sector spansH over a reducible ⊕ representation. The adjoint and the spinor multiplets contain three SM singlets that may acquire GUT scale VEVs. 45 16 45 16 As we have seen in Chapter 2 the phenomenologically favored scenarios allowed by gauge coupling unification correspond to a three-step breaking along one of the following directions: MU MI MB L − SO C L R B L C L R B L ; ÊÏ − ÊÏ − ÊÏ SM (3.1) MU MI MB L − SO(10) 3C 2L 2R 1 C L3 R2 B1L 1 ; ÊÏ ÊÏ − ÊÏ SM (3.2) (10) 4 2 1 MU 3 2 M1 I 1 H where the first two breaking stages at and MareB L driven by the VEVs,H while the breaking to the SM at the intermediate scale − is controlled by the . The MU MI >45 MB L constraints coming from gauge unification are such that ≫ − . In par- SU 16 ticular, even without proton decay limits, any intermediate -symmetricH stage isH excluded. On the other hand, a series of studies in the early 1980’s of the ⊕ (5) model [56, 57, 58, 59] indicated thatSU the onlyU intermediate stagesH allowed by the scalar ⊗ 45 16 sectorSU dynamics were the flippedH for leading VEVs orSO the standard GUT for dominant VEV. This observation excluded the simplest Higgs (5) (1) 45 sector from realistic consideration. (5) 16 (10) In this chapter we show that the exclusion of the breaking patterns in Eqs. (3.1)– (3.2) is an artifact of the tree level potential. As a matter of fact, some entries of the Chapter 3. The quantum vacuum of the minimal SO GUT 96 (10) scalar hessian are accidentally over-constrained at the tree level. A number of scalar interactions that, by a simple inspection of the relevant global symmetries and their explicit breaking, are expected to contribute to these critical entries, are not effective at the tree level. On the other hand, once quantum corrections are considered, contributions of O MU / π induced on these entries open in a natural way all group-theoretically 2 2 allowed vacuum configurations. Remarkably enough, the study of the one-loop effective ( 16 ) potential can be consistently carried out just for the critical tree level hessian entries (that correspond to specific pseudo-Goldstone boson masses). For all other states in the scalar spectrum, quantum corrections remain perturbations of the tree level results and do not affect the discussion of the vacuum pattern.

Let us emphasizeSO that the issue we shall be dealingH with is inherent to all nonsupersymmetric models with one adjoint governing the first breaking step. Only one additional scalar representation interacting with the adjoint is sufficient to (10) 45 SO demonstrate conclusively our claim. In this respect, the choice of the spinor to trigger the intermediate symmetry breakdown is a mere convenience and a similar B L (10) line of reasoning can beSO devised for the scenarios in which − is broken for instance by a 126-dimensional tensor. SO We shall therefore study the structure of the vacua of a Higgs potential with H H (10) only the ⊕ representation at play. Following the common convention, we define H χ χ χ (10) ≡ and denote by and − the multiplets transforming as positive and negative 45 16 SO chirality components of+ the reducible 32-dimensional spinor representation 16 φ respectively. Similarly, we shall use the symbol (or the derived for the components (10) H in the natural basis, cf. Appendix B) for the adjoint Higgs representation . Φ 3.1.1 The tree-level Higgs potential 45

The mostH generalH renormalizable tree-level scalar potential which can be constructed out of and reads (see for instance Refs. [45, 195]): 45 16 V V Vχ V χ ; (3.3) 0 Φ Φ = + + where, according to the notation in Appendix B, µ a a V ; − 2 Tr Tr Tr (3.4) 2 1 2 2 2 4 Φ ν λ λ Vχ = χ†χΦ + χ†(χ Φ ) +χ† jχ Φ χ† j χ − 22 4 4 − − 1 2 2 + + = + ( ) + ( Γ )( Γ ) and 2 4 4 V χ α χ†χ βχ† χ τχ† χ: Tr (3.5) 2 2 Φ = ( ) Φ + Φ + Φ 3.1. The minimal SO(10) Higgs sector 97

The mass terms and coupling constants above areSO real by hermiticity. The cubic self-interaction is absent due the zero trace of the adjoint representation. For Φ the sake of simplicity, all tensorial indices have been suppressed. (10) 3.1.2 The symmetry breaking patterns The SM singlets

H H SO There are in general three SM singlets in the ⊕ representation of . C L R B L U B L Labeling the field components according to − (where the − generator B L / ; ; ;45 16; ; ; (10)H is − ), the SM singlets reside in the and submultiplets of ; ; ; H 3 2 2 1 (1) and in the component of . We denote their VEVs as ( ) 2 1 (1 1 1 0) (1 1 3 0) 45 2 ; ; ; ωB L; (1 1 2 + ) h 16 i ≡ − ; ; ; ωR; h(1 1 1 0)i ≡ (3.6) ; ; ; χR; (1 1 3 0) ≡ 1 ωB L;R χR 2 H − (1 1 2 + ) where are real and can be taken real by a phase redefinitionSO of the . Dif- ferent VEV configurations trigger the spontaneous breakdown of the symmetry χR 16 into a number of subgroups. Namely, for one finds (10) ωR ; ωB L C L R B L − 6 =0 − ωR ; ωB L C L R 6=0 − =0: 3 2 2 1 ωR ; ωB L C L R B L 6=0 − 6=0: 4 2 1− (3.7) ωR ωB L ′ Z′ =0− − 6 =0:flipped 3 2 1 1 ωR ωB L Z = − 6 =0 : standard5 1 SU U Z ′ Z′ = =0 : 51 with andSO standing for the two different embedding of the ⊗ subgroup into , i.e. standard and “flipped” respectively (see the discussion at the end 51 5 1 (5) (1) of the section). χR (10) When 6 all intermediate gauge symmetries are spontaneously broken downSU to the SM group, with the exception of the last case which maintains the standard =0 subgroup unbroken and will no further be considered. (5) The classification in Eq. (3.7) dependsH on theH phase conventions usedωR inωB theL parametriza- ⊕ − tion of theSU SM singlet subspaceω ofR ωB L . The statement that yields the standard vacuum while − − corresponds to the flipped setting defines 45 16 = a particular basis in this subspace (see Sect. 3.1.2). The consistency of any chosen (5) = framework is then verified against the corresponding Goldstone boson spectrum. H H SOThe decomposition of the and representations with respect to the relevant subgroups is detailed in Tables 4.4 and 4.5. 45 16 (10) Chapter 3. The quantum vacuum of the minimal SO GUT 98 (10)

C L R C L R C L R B L C L R B L C L Y ′ Z Y − − ′ ′ ; ; ; ; ; ; ; ; ; ; ; ; ; 4 2 2 4 2 1 3 ;2 ; 2; 1 1 3 ;2 ; 1; 1 1 3 ;2 ; 1 1 5 5; 1 1 1 − 6 − 6 − 6 − − 6 (4 ;2 ;1) (4 ;2 ;0) 3; 2; 1; + 1 3; 2; 0 +; 1 3; 2; + 1 10 (10; +1) + 1 − 2 −2 2 − 2 ; ; 1 1 2 1 1 1; 2; 0 1 ; 1 1; 2; 1 5 5; 3 2 − 2 6 − 2 − 6 − 3 − 3 4 1 2 4 1 + 1 3; 1; 2; 3; 1; + 1 ; 1 3; 1; + 2 5 (10; +1) 1 2 2 6 3 3 4 1 1 3; 1; 1 ; 1 3; 1; 10 5 ; 3 + 2 − 2 2 1 1 2 + 1 1 + 1 + 1 (1 1 +1) 10 (1 +5) 0 2 2 SO Table 3.1: Decomposition of the spinorial representation1 1 + with(1 1 respect0) to1 the various(10 +1) +1sub- groups. The deﬁnitions and normalization of the abelian charges16 are given in the text. (10)

C L R C L R C L R B L C L R B L C L Y ′ Z Y − − ′ ′ ; ; ; ; ; ; ; ; ; ; ; ; ; − 4 2 2 4 ; 2; 1 3 2 2 1 3 ; 2; 1; 1 3 ; 2; 1 5 5 ;1 1 (1 1 3) (1; 1; +1) (1 1 3 0) (1; 1; +1; 0) (1; 1; +1) 10 (10; 4) +1 (1 1 0−) (1 1 −0 0) (1 1 −0) 1 (1 0) −0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; (1 ;1 ; 1) ; ; ; (1 ;1 ; 1 ;0) (1 ;1 ; 1) 10 10 ;+4 1 − − − (1 3 1) (1 ;3 ;0) 1 (1 3 1 0) 1 (1 ;3 ;0 01 ;) 1 (1 ;3 ;01) 24 (24; 0) 05 − 2 3 − 2 − 3 −6 − 6 (6 2 2) 6 2 + 1 3; 2; 2; 3; 2; + 1 ; 1 3; 2; 5 10 (24; 0) 1 2 2 3 6 − 6 6 2 1 3; 2; 1 ; 1 3; 2; 5 24 (10 ; 4) + 1 3 − 2 3 − 6 6 ; ; ; ; 3; 2; 2; + 3; 2; +; 1 + 1 3; 2; + 1 24 10 ;+4 5 2 3 6 6 ; ; ; 3; 2; ; + 3; 2; 10 (24; 0) + (15 1 1) (15 1 0) (1 ;1 ;1 ;0) 2 (1 ;1 ;0 ;0) 2 (1 ;1 ;0) 2 24 (24; 0) 02 − 3 − 3 − 3 − − 3 3; 1; 1; + 2 3; 1; 0; + 2 3; 1; + 2 10 10 ;+4 + 2 3 3 3 3 3 1 1 3 1 0 3 1 10 (10 4) SO Table 3.2: Same(8 1 as1 in0) Table 4.4(8 for1 the0 0) adjoint(8 1 0 ( ) ) representation.24 (24 0) 0 (10) 45 3.1. The minimal SO(10) Higgs sector 99

The L-R chains

According to the analysis in Chapter 2, the potentiallySO viable breaking chains fulﬁlling the basic gauge uniﬁcation constraints (with a minimal Higgs sector) correspond to the settings: (10)

ωB L ωR >χR SO C L R B L C L R B L C L Y − ≫ → − → − → (3.8) : (10) 3 2 2 1 3 2 1 1 3 2 1 and ωR ωB L >χR SO C L R C L R B L C L Y : ≫ − → → − → (3.9)

: (10)χR 4ω2R 1 χR 3 2ωB1 L1 3 2 1 AsSO remarked in Sect. 2.3.2, the cases ∼ or ∼ − lead to eﬀective two-step breaking patterns with a non-minimal set of surviving scalars at the intermediate scale. On the other hand, a truly two-step setup can be recovered (with a minimal (10) ωR ωB L ﬁne tuning) by considering the cases where or − exactly vanish. Only the explicit study of the scalar potential determines which of the textures are allowed.

Standard SU(5) versus ﬂipped SU(5)

SU SO There are in general two distinct SM-compatibleSU embeddings of into [69, 70]. They differ in one generator of the Cartan algebra and therefore in the U Z (5) (10) cofactor. (5) Y TR TB L In the “standard” embedding, the weak hypercharge operator − belongs (1) SU Z 3 Ti;Z to the algebra and the orthogonal Cartan generator (obeying for Ti SU Z TR TB L = + all ∈ ) is given by − − . (5) SU ′ 3 [ ]=0 In the “flipped”(5) SU ′ case,= the4 right-handed+6 isospin assignment of quark and leptons into the Y multipletsT T is turned over so that the “flipped”U hypercharge gen- ′ (5)R B L Z′ erator reads − − . Accordingly, the additional generator reads Z′ TR TB L(5) 3 Ti;Z′ Ti SU ′ − , such that for all ∈ . Weak hypercharge is then 3 Y Z′ = Y ′ / + (1) given by − . =4 +6 [ ]=0 SU (5) Tables 4.4–4.5=SO ( show) 5 the standard and flipped decompositions of the spinorial and adjoint representations respectively. SU (5) The two (10)vacuaSU in Eq. (3.7) differ by the texture of theZ adjoint representation VEVs: in the standard case they are aligned with the operator while they Z′ (5) SU ′ match the structure in the flipped setting (see Appendix B.4 for an explicit (5) representation). (5) Chapter 3. The quantum vacuum of the minimal SO GUT 100 3.2 The classical vacuum (10)

3.2.1 The stationarity conditions

By substituting Eq. (3.6) into Eq. (3.3) the vacuum manifold reads

V µ ωR ωB L a ωR ωB L h i − − − a 2 2 2 2 2ν 2 λ 0 1 ωR ωB L ωRωB L χR χR αχR ωR ωB L = 2 (2 +3 − )+4 (2 −+3− 2 ) − 2 4 4 2 2 β 2 1 4 τ 2 2 2 + (8 + 21 + 36 ) χR ωR + ωB L+4 χ(2R ωR+3 ωB )L − − − (3.10) 4 2 2 4 2 2 + (2 +3 ) (2 +3 ) The corresponding three stationary conditions4 can be conveniently2 written as ∂ V ∂ V ∂ V ∂ V ∂ V h i h i ; ωB L h i ωR h i ; h i ; ∂ωR − ∂ωB L − ∂ωR − ∂ωB L ∂χR (3.11) 0 −0 0 −0 0 1 2 2 =0 =0 =0 which8 lead respectively3 to 3 a µ a ωR ωB L ωR ωB L ωB LωR αχR − − − − − 2 2 2 2 2 2 2 1 ωR ωB L ; [ +4 (2 +3 ) + (4 +7 2 )+2× ] − − (3.12) 4 ( )=0 a ωR ωB L ωRωB L βχR ωR ωB L τχR ωR ωB L ; − − − − − − − (3.13) 2 2 2 β [ ν4 ( λ χ+R α) ωR ωB L (2 +3ωR ω)B + L ](τ ωR ω)=0B L χR : − − − − − (3.14) 2 2 2 2 2 1 [ + +8 (2 +3 ) + (2 +3 ) (2 +3 )] =0 SU We have chosen linear combinations that factor out the uninteresting standard ⊗ U Z ωR ωB L 2 solution, namely − . χR (5) In summary, when , Eqs. (3.12)–(3.13) allow for four possible vacua: (1) = ω ωR ωB L Z • − (standard=0 ) ω ωR ωB L ′ Z • = = − − (flipped 51 ′ ) ωR ωB L C L R B L • = and= − 6 ( 5 1 − ) ωR ωB L C L R • =06 and − =0 (3 2 2 ) 1 tree level As we=0 shall see, the=0 last4 two2 options1 are not minima. Let us remark that χR H H for 6 , Eq. (3.13) impliesβ naturallyτ a correlation among the and VEVs, or a fine tuned relation between and , depending on the stationary solution. In the cases ωR =0ωB L ωR ωB L τ βω τ 45βωB L 16 τ βωR − − , and − one obtains , − and respectively. Consistency with the scalar mass spectrum must be verified in each case. = = 0 = 0 = = 3 = 2 3.2. The classical vacuum 101 3.2.2 The tree-level spectrum

The gauge and scalar spectra corresponding to the SM vacuum conﬁguration (with H H non-vanishing VEVs in ⊕ ) are detailed in Appendix D. The scalar spectra obtained in various limits of the tree-level Higgs potential, corre- 45 16 sponding to the appearance of accidental global symmetries, are derived in Apps. D.2.1– χR D.2.5. The emblematic case is scrutinized in Appendix D.2.6. 3.2.3 Constraints on the=0 potential parameters

The parameters (couplings and VEVs) of the scalar potential are constrained by the requirements of boundedness and the absence of tachyonic states, ensuring that the vacuum is stable and the stationary points correspond to physical minima.

NecessaryχR conditions for vacuum stability are derived in Appendix C. In particular, on the section one obtains a > a : =0 − (3.15) 13 1 80 2 Considering the general case, the absence of tachyons in the scalar spectrum yields among else a < ; <ωB L/ωR < : − − − (3.16) 1 ωB L2/ωR − 2 The strict constraint on ; ; 0is a consequence; 2; of the tightlyH correlated form of the tree-levelC massesL Y of the and submultiplets of , labeled according to the SM ( ) quantum numbers, namely (8 1 0) (1 3 0) 45 M ; ; a ωB L ωR ωB L ωR ; 3 2 1 − − − (3.17) 2 M ; ; a2 ωR ωB L ωR ωB L ; (1 3 0)=2 ( − − )( +2 − ) (3.18) 2 2 (8 1 0)=2 ( )( +2 ) that are simultaneously positive only if Eq.τ (3.16) is enforced. For comparison withZ previous studies, let us remark that in the limit (corresponding to an extra symmetry → − ) the intersection of the constraints from Eq. (3.13), Eqs. (3.17)–(3.18)2 ; ; =; 0; / and the mass eigenvalues of the and states, yields Φ Φ a < ; ωB L/ωR ; (1 1 −1) ≤ (3− 2 1 6)≤ − (3.19) 2 2 3 thus recovering the results of Refs.0 [56,1 57, 58, 59].

SU In′ eitherU Z case, oneχR concludes by inspecting the scalar mass spectrum that flipped ⊗ ′ is for the only solution admitted by Eq. (3.13) consistent with the χR constraints in Eq. (3.16) (or Eq. (3.19)). For 6 , the fine tuned possibility of having ω(5)B L/ωR(1) =0 χR or − ∼ − such that is obtained at an intermediate scale fails to reproduce =0 the SM couplings (see e.g. Sect. 2.3.2). Analogous and obvious conclusions hold for ωB L ωR χR 1 MU χR ωR;B L SU − ∼ ∼ ∼ and for ≫ − (standard in the first stage). (5) Chapter 3. The quantum vacuum of the minimal SO GUT 102

SO(10) This is the origin of the common knowledge that nonsupersymmetric settings with the adjoint VEVs driving the gauge symmetry breaking are not phenomeno- H (10) logically viable. In particular, a large hierarchy between the VEVs, that would set the stage for consistent uniﬁcation patterns, is excluded. 45 The key question is: why are the masses of thea states in Eqs. (3.17)–(3.18) so tightly correlated? Equivalently, why do they depend on only? 2 3.3 Understanding the scalar spectrum

A detailed comprehension of the patterns in the scalar spectrum may be achieved by understanding the correlations between mass textures and the symmetries of the scalar potential. In particular, the appearance of accidental global symmetries in limiting cases may provide the rationale for the dependence of mass eigenvalues from speciﬁc couplings. To this end we classify the most interesting cases, providing a counting of the would-be Goldstone bosons (WGB) and pseudo Goldstone bosons (PGB) for each case. A side beneﬁt of this discussion is a consistency check of the explicit form of the mass spectra.

3.3.1 45 only with a

H V Let us first consider the potential2 generated by , namely in Eq. (3.3). When a =H 0 , i.e. when only trivial invariants (built off moduli) are considered,Φ the scalar 45O potential2 exhibits an enhanced global symmetry: . The spontaneous symmetry =0 45 H O breaking (SSB) triggered by the VEV reduces the global symmetry to . As a (45) consequence, 44 massless states are expected in the scalar spectrum. This is verified 45 SO (44) explicitly in Appendix D.2.1. Considering the case of the gauge symmetry SU ′ U Z′ broken toSO the flipped/SU U⊗ , − WGB, with the quantum numbers of ′ Z′ (10) the coset ⊗ algebra, decouple from the physical spectrum while, (5) (1) 45 25 = 20 a − PGB remain, whose mass depends on the explicit breaking term . (10) (5) (1) 2 44 20 = 24 3.3.2 16 only with λ

λ Vχ We proceed in analogy with2 the previous discussion. Taking in enhances the O = 0 O O H global symmetry to . The spontaneous breaking of 2 to due to the =0 VEV leads to 31 massless modes, as it is explicitly seen in Appendix D.2.2. Since the SO (32) χR SU(32) (31) 16 gauge symmetry is broken bySO to/SU the standard , − WGB, with the quantum numbers of the coset algebra, decouple from the physical (10) (5) 45 24 = 21 spectrum, while − PGB do remain. Their masses depend on the explicit λ (10) (5) breaking term . 31 21 = 10 2 3.3. Understanding the scalar spectrum 103 3.3.3 A trivial 45-16 potential a λ β τ

2 2 H H When only trivial invariantsV (i.e. moduli)O( = ofO both= = and= 0) are considered, the global symmetry of in Eq. (3.3) is ⊗ . This symmetry is spontaneously O O H H 45 16 broken into ⊗ 0 by the and VEVs yielding 44+31=75 GB in the (45) (32) SO scalar spectrum (see Appendix D.2.4). Since in this case, the gauge symmetry is (44) (31) 45 16 broken to the SM gauge group, − WGB, with the quantum numbers of the SO /SM (10) coset algebra, decouple from the physical spectrum, while − 45 12 = 33 PGB remain. Their masses are generally expected to receive contributions from the (10) a λ β τ 75 33 = 42 explicitly breaking terms , , and . 2 2 3.3.4 A trivial 45-16 interaction β τ β τ Turning off justχ the and couplings( still= allows= 0) for independent global rotations of the a and λ Higgs fields.V The largestO globalO symmetries are those determined by the and terms in , namely SU andU , respectively. ConsiderSU the Φ ′ Z′ spontaneous2 breaking2 to global0 flipped 45 ⊗ and16 the standard by the H H (10) (10) and VEVs, respectively. This setting gives rise to massless scalar SO (5) (1) (5) modes. The gauged symmetry is broken to the SM group so that 33 WGB 45 16 20+21 = 41 decouple from the physical spectrum. Therefore, 41-33=8 PGB remain, whose masses (10) β τ receive contributions from the explicit breaking terms and . All of these features are readily verified by inspection of the scalar mass spectrum in Appendix D.2.5. 3.3.5 A tree-level accident

; ; ; ; H The tree-level masses of the cruciala not and multiplets belonging to the depend only on the parameter but on the other parameters expected (cf. 3.3.3), λ β τ (1 3 0) (8 1 0) 45 namely , and . 2 λ τ H While2 the and terms cannot obviously contribute at the tree level to mass β χR terms, one would2 generally expect a contribution from the term, proportional to . σij φij / σij i;j ;::; i j 45 2 Using the parametrizationSO , where the ( ∈ { }, 6 ) matrices represent the algebra on the 16-dimensional spinor basis (cf. Appendix B), one H Φ = 4 1 10 = obtains a mass term of the form (10) β 45 χR σij β σkl β φij φkl : (3.20) 2 16 16 φij ( ) ( ; ); ; ; The projection of the fields16 onto the and components lead, as we know, to vanishing contributions. (1 3 0) (8 1 0) This result can actually be understood on general grounds by observing that the scalar interaction in Eq. (3.20) has the same structureH as the gauge boson mass from the covariant-derivative interaction with the , cf. Eq. (D.7). As a consequence, no 16 Chapter 3. The quantum vacuum of the minimal SO GUT 104 (10) β H tree-level mass contribution from the couplingSU can be generated for the scalars carrying the quantum numbers of the standard algebra. This behavior can be 45 again verified by inspecting the relevant scalar spectra in Appendix D.2. (5) The above considerations provide a clear rationale for the accidental tree level ωB L/ωR χR constraint on − , that holds independently on the size of . β τ O MU / π On the other hand, we; should; expect; ; the and interactions to contribute terms to the masses of and at the quantum level. Similar contributions ( 4 ) should also arise from the gauge interactions, that break explicitly the independent (1 3 0) H (8 1 0)H global transformations on the and discussed in the previous subsections. H The typical one-loop self energies, proportional to the VEVs, are diagrammati- 45 16 H χR cally depicted in Fig. 3.1. While the exchange of components is crucial, the is not 45 needed to obtain the large mass shifts. In the phenomenologically allowed unification 16 patterns it gives actually negligible contributions.τ It is interesting to notice that the -inducedSO mass corrections do not depend on the gauge symmetry breaking, yielding an symmetric contribution to all scalars in H . (10) 45 χ φ φ τ τ χ hφi hφi χ φ φ β β χ hφi hφi

φ φ g2 g2

χ O τ/ π; β φ / π; g φ / π Typical one-loop diagrams that induce for h i , h i h i renormal- Figure 3.1: H 2 ization to the mass of ﬁelds ata the uniﬁcation scale. They= 0 are( relevant4 for4 the PGB4 states) , whose tree level mass is proportional45 to . 2 One is thus lead to the conclusion that any result based on the particular shape of the 3.4. The quantum vacuum 105

H tree-level vacuum is drastically aﬀectedτ at the quantum level. Let us emZphasize that although one may in principle avoid the -term by means of e.g. an extra symmetry, 45 β no symmetry can forbid the -term and the gauge loop contributions. 2 H H B L In case one resorts to , in place of , for the purpose of − breaking, H† H H the more complex tensor structure of the class of quartic invariants in 126 16 2 ; ; ; ; the scalar potential may admit tree-level contributions to the states and H 126 45 126 H proportional to h i. On the other hand, as mentioned above, whenever h i is (1 3 0) H (8 1 0) small on the uniﬁcation scale, the same considerations apply, as for the case. 126 126 16 3.3.6 The χR limit

FromSU the previous= discussion 0 it is clear that the answer to the question whether the non- vacua are allowed at the quantum level is independent on the speciﬁc value B L χR MU of the − breaking VEV ( ≪ in potentially realistic cases). (5) In order to simplify the study of the scalar potentialχR beyond the classical level it is therefore convenient (and suﬃcient) to consider the limit. χR H H When the mass matrices of the and sectors are not coupled. The =0 stationary equations in Eqs. (3.12)–(3.13) lead to the four solutions = 0 45 16 ω ωR ωB L Z • − ( ) ω ωR ωB L ′ Z • = = − − 51( ′ ) ωR ωB L C L R B L • = and= − 6 5 1 ( − ) ωR ωB L C L R • 6=0 and − =0 (3 2 2 ) 1

In what follows,=0 we will=0 focus4 our2 1 discussion on the last three cases only. χR It is worth noting that the tree level spectrum in the limit is not directly obtained from the general formulae given in Appendix D.2.3, since Eq. (3.14) is trivially χR = 0 satisﬁed for . The corresponding scalar mass spectraSU are derived and discussed in Appendix D.2.6. Yet again, it is apparent that the non vacuum conﬁgurations =0 exhibit unavoidable tachyonic states in the scalar spectrum. (5) 3.4 The quantum vacuum

3.4.1 The one-loop eﬀective potential

We shall compute the relevant one-loop corrections to the tree level results by means of the one-loop effective potential (effective action at zero momentum) [196]. We can formally write V V Vs Vf Vg ; (3.21) eff 0 = + ∆ + ∆ + ∆ Chapter 3. The quantum vacuum of the minimal SO GUT 106 (10) V Vs;f;g where is the tree level potential and denote the quantum contributions induced by0 scalars, fermions and gauge bosons respectively. In dimensional regulariza- ∆ MS tion with the modified minimal subtraction ( ) and in the Landau gauge, they are given by η W φ;χ Vs φ;χ;µ W φ;χ ; π Tr 2µ − (3.22) 4 κ2 M (φ;χ2 ) 3 ∆Vf (φ;χ;µ) = − M (φ;χ ) log ; 64π Tr 2µ − 2 (3.23) 4 2 ( 2φ;χ) 3 ∆Vg(φ;χ;µ) = ( φ;χ) log M ; 64π Tr M 2µ −2 (3.24) 4 3 ( ) 5 η ∆ ( ) = 2 ( κ ) log 2 W M with for real (complex)64 scalars and for Weyl (Dirac)6 fermions. , and M are the functional scalar, fermion and gauge boson mass matrices respectively, = 1(2) = 2(4) as obtained from the tree level potential. W φ;χ In the case at hand, we may write the functional scalar mass matrix, as a 2 77-dimensional hermitian matrix, with a lagrangian term ( ) ψ†W ψ; (3.25) 2 1 ψ φ;χ;χ∗ W defined on the vector basis 2 . More explicitly, takes the block form 2 V V V = ( ) φφ φχ φχ∗ W φ;χ  Vχ φ Vχ χ Vχ χ  ; ∗ ∗ ∗ ∗ (3.26) 2 V V V  χφ χχ χχ∗    ( ) =   where the subscriptsφ χ χ denote∗ the derivatives of the scalar potential withV respectV to the set of fields , and . In the one-loop part of the effective potential ≡ . We neglect the fermionic component of the effective potential since there0 are no fermions at the GUT scale (we assume that the right-handed (RH) neutrino mass is substantially lower than the unification scale). φ;χ The functional gauge boson mass matrix, M is given in Appendix D, Eqs. (D.6)– 2 (D.7). ( ) 3.4.2 The one-loop stationary equations

The first derivative of theψa one-loop part of the effective potential, with respect to the scalar field component , reads ∂ V W s W ;W W W ψa ψa ∂ψa π Tr µ 2 − (3.27) 2 2 2 2 ∆ 1 3 = 2 log 2 + 64 2 3.4. The quantum vacuum 107

Wψ W ψa where the symbol a stands for the partial derivative of with respect to . Anal- 2 ∂ Vf; g /∂ψa 2 ogous formulae hold for W . The trace properties ensure that Eq. (3.27) holds independently on whether does commute with its first derivatives or not. ∆ 2 The calculation of the loop corrected stationaryχR equationsH dueH to gauge bosons and scalar exchange is straightforward (for the and blocks decouple in Eq. (3.26)). On the other hand, the corrected equations are quite cumbersome and we = 0 45 16 do not explicitly report them here. It is enough to say that the quantum analogue of Eq. (3.13) admits analytically the same solutions as we had at the tree level. Namely, ωR ωB L ωR ωB L ωR ωB L − − − − these are Z , ′ Z C, L R B Land C L R , corresponding respectively to the standard , flipped ′ , − and preserved subalgebras. = = =0 =0 3.4.3 The51 one-loop5 scalar1 3 2 mass2 1 4 2 1 V In order to calculate the second derivatives of the one-loop contributionsW to it is in general necessary to take into account the commutation properties of witheff its 2 derivatives that enter as a series of nested commutators. The general expression can be written as ∂ V W s W W W W W ;W W ;W ψa ψb ψaψb ψaψb ψa ψb ∂ψ2a∂ψb π µ 2 − h 2 2 2 2 hn 2 2o n 2 2 oi ∆ 1 2 m 2 3 = ∞Tr + m + + log m k 64 m W ;W W ; :: W ;W :: W − 2 − m k ψa ψb − m k X +1 1 X n 2 2 oh 2 h 2 2 i i 2 i + ( 1) 1 (3.28) =1 =1 k where the commutators in the last line are taken − times. Let us also remark that, although not apparent, the RHS of Eq. (3.28) can be shown to be symmetric under a b 1 ↔ , as it should be. In specific cases (for instance when the nested commutatorsW vanish or they can be rewritten as powers of a certain matrix commuting with ) the functional mass evaluated on the vacuum may take a closed form. Running and pole mass

The effective potential is a functional computed at zero external momenta. Whereas the stationary equations allow for the localization of the new minimum (being the VEVs translationally invariant), the mass shifts obtained from Eq. (3.28) define the running mab masses 2 ∂ V φ mab mab ab ≡ ∂ψ2 a∂ψb ψ (3.29) 2 eff h i 2 ( ) mab = ab p+Σ (0) MS where are the renormalized masses and are the renormalized self- 2 Ma 2 energies. The physical (pole) masses are then obtained as a solution to the equation 2 Σ ( ) p δab mab ab p det − (3.30) 2 2 2 + ∆Σ ( ) =0 Chapter 3. The quantum vacuum of the minimal SO GUT 108 (10) where ab p ab p ab − (3.31) 2 2 For a given eigenvalue ∆ΣM(a )=Σma ( )a MΣa (0) (3.32) 2 2 2 gives the physical mass. The gauge= and scheme+ ∆Σ ( dependence) in Eq. (3.29) is canceled by the relevant contributions from Eq. (3.31). In particular, infrared divergent terms in Eq. (3.29) related to the presence of massless WGB in the Landau gauge cancel in

Eq. (3.32). Ma Of particular relevance is the case when is substantially smaller than the (GUT- µ MU Ma MU scale) mass of the particles that contribute to . At , in the ≪ limit, 2 2 one has a Ma O MΣ(0)a/MU : = (3.33) 2 4 2 In this case the running mass∆Σ computed( ) = from( Eq. (3.29)) contains the leading gauge independent corrections. As a matter of fact, in order to study the vacua of the potential in Eq. (3.21), we need to compute the zero momentum mass corrections just to those states that areM tachyonicU / π at the tree level and whose corrected mass turns out to be of the order of . We may safely neglect the one loop corrections for all other states with masses of MU 4 χR order . It is remarkable, as we shall see, that for the relevant corrections to the masses of the critical PGB states can be obtained from Eq. (3.28) with vanishing = 0 commutators. 3.4.4 One-loop PGB masses

ωB L/ωR The stringent; ; tree-level; ; constraint on the ratio − , coming from the positivity of the and masses,a follows from the fact that some scalar masses depend only on the parameter . On the other hand, the discussion on the would-be global (1 3 0) (8 1 0) symmetries of the scalar2 potential shows that inτ generalβ their mass should depend on other terms in the scalar potential, in particular andO φ. / π h i A set of typicalH one-loop diagrams contributing renormalizationH to the masses of states is depicted in Fig. 3.1. As we already pointed out the VEV does ( 4 ) H not play any role in the leading GUT scale corrections (just the interaction between H 45 16 and , or with the massive gauge bosons is needed). Therefore we henceforth work χR 45 in the strict limit, that simplifies substantially the calculation. In this limit the 16 scalar mass matrix in Eq. (3.26) is block diagonal (cf. Appendix D.2.6)V and the leading = 0 χ∗χ corrections from the one-loop effective potential are encoded in the sector. H More precisely, we are interesteda in the corrections to those scalar states whose tree level mass depends only on and have the quantum numbers of the preserved 45 non-abelian algebra (see Sect. 3.3.12 and AppendixW D.2.6). It turns out that focusing to this set of PGB states the functional mass matrix and its first derivative do commute 2 3.4. The quantum vacuum 109

χR for and Eq. (3.28) simplifies accordingly. This allows us to compute the relevant mass corrections in a closed form. =0 ; ; The calculation of the EP running mass from Eq. (3.28) leads for the states ; ; µ MU and at to the mass shifts M ; ; (1 3 0) (8 1 0) = 2 τ β ωR ωRωB L ωB L g ωR ωB LωR ωB L − − − − − ; ∆ (1 3 0) = 2 2 2 2 π 4 2 2 (3.34) + (2 +2 ) + 16 + + 19 M ; ; 2 4 2 τ β ωR ωRωB L ωB L g ωR ωB LωR ωB L − − − − − ; ∆ (8 1 0) = 2 2 2 2 π 4 2 2 (3.35) + ( +3 ) +2 13 + + 22 where the sub-leading (and gauge dependent)4 logarithmic terms are not explicitly reported. For the vacuum configurations of interest we find the results reported in Appendix E. In particular, we obtain ω ωR ωB L ′ Z • − − ( ′ ): τ β g ω = = M5 1 ; a ω ; − 2 2 π 4 2 (3.36) 2 2 2 + (5 + 34 ) ωR ωB L (24C0)L =R B4 L + 2 • and − 6 ( − ): 4 τ β g ωB L =0M ; ; ; =0M 3 ;2 ;2 ;1 a ωB L − ; − 2 2 π 4 2 (3.37) 2 2 2 τ 2 β g +ω (2B L + 192 ) M (1; 3; 1; 0) = a(1ω1B 3L 0)=2 + − ; − − 2 2 π 4 2 4 (3.38) 2 2 2 + (3 + 22 ) ωR (8 1ωB1 L0) = 4 C L R + 2 • 6 and − ( ): 4 τ β g ωR =0 =0M ;4 ;2 1 a ωR ; − 2 2 π 4 2 (3.39) 2 2 2 τ + (2β +2 16g ω)R M (1 3; 0); = 4a ωR + : 2 2 4π 4 2 (3.40) 2 2 2 + ( + 13 ) (15 1 0)=2 + 2 In the effective theory language Eqs. (3.36)–(3.40) can4 be inSOterpreted/G as the one-loopG GUT-scale matching due to the decoupling of the massive states where is the preserved gauge group. These are the only relevant one-loop corrections needed (10) in order to discuss the vacuum structure of the model. It is quite apparent that a consistent scalar mass spectrum can be obtained in all cases, at variance with the tree level result. SU In order to fully establish the existence of the non- minima at the quantum level one should identify the regions of the parameter space supporting the desired (5) vacuum configurations and estimate their depths. We shall address these issues in the next section. Chapter 3. The quantum vacuum of the minimal SO GUT 110 3.4.5 The one-loop vacuum structure (10) Existence of the new vacuum configurations

The existence of the different minima of the one-loop effective potential is related to a β τ g µ MU ′ Z′ the values of the parameters , , and at the scale . For thea flipped< case it is sufficient, as one expects,2 to assume the tree level condition . On the = 5 1 other hand, from Eqs. (3.37)–(3.40) we obtain 2 0 ωR ωB L C L R B L • and − 6 ( − ): τ =0 =0 3 2 2π 1a < β g ; − ωB2 L (3.41) 2 − 2 4 2 8 2 +2 + 19 ωR ωB L C L R • 6 and − ( ): τ =0 =0 4 2 1 π a < β g : − ωR2 (3.42) 2 2 4 2 8 2 + + 13 τ ωY;R a < − Considering for naturalness ∼ , Eqs. (3.41)–(3.42) imply | | . This con- 2 straint remains within the natural perturbative range for dimensionless2 couplings. a 10 While all PGB states whose mass is proportional to − receive large positive loop corrections, quantum corrections are numerically irrelev2ant for all of the states witha GUT scale mass. On the same groundsH we may safely neglect the multiplicative loop corrections induced by the states on the PGB masses. 2 Absolute minimum 45 SU It remains to show that the non solutions may actually be absolute minima of the potential. To this end it is necessary to consider the one-loop corrected stationary (5) equations and calculate the vacuum energies in the relevant cases. Studying the shape of the one-loop effective potential is a numerical task. On the other hand, in the approximation of neglecting at the GUT scale the logarithmic corrections, we may reach non-detailed but definite conclusions. For the three relevant vacuum configurations we obtain: ω ωR ωB L ′ Z • − − ( ′ ) ν αν βν τ V=ω; χR= 5 1 ω − π π π − π 4 2 2 2 3 5 5 5 2 ( =0)= a 2 + a 2 +a a 2 a2 α αβ β g (3.43) a 16 16 16 ω ; − − π 2 − π − π 2 π 2 π π2 − π4 2 1 1 2 2 4 1 65 600 45 645 100 25 65 5 + 100 + 2 2 2 + 2 + 2 + 2 2 4 32 2 64 2 3.4. The quantum vacuum 111

ωR ωB L C L R B L • and − 6 ( − ) ν αν βν τ V =0ωB L; χR =0 3 2 2 1 ωB L − − π π π − π 4 2 2 2 − 3 3 3 3 2 ( =0)=a 2a+ 2a a+ 2 a 2α αβ β g (3.44) a 16 16 16 ωB L ; − − π 2 π π2 π 2 π π2 − π4 − 2 1 1 2 2 4 1 21 216 33 45 36 9 21 15 + 36 + 2 + 2 + 2 + 2 + 2 + 2 2 ωR ωB L4 C L R 32 2 64 16 • 6 and − ( ) ν αν βν τ =0V ωR; χR =0 4 2 1 ω − π π π − π R 4 2 2 2 3 2 2 ( =0)= 2a+ a2 a+ 2 a 2 α αβ β g (3.45) a a 16 8 8 ωR : − − π 2 π π 2 π 2 π π2 − π4 1 1 2 2 4 1 2 96 42 147 16 2 7 + 16 2 + 2 + 2 + 2 + 2 + 2 + 2 2 32 8 16 A simple numerical analysis reveals that for natural values of the dimensionless couplings and GUT mass parameters any of the qualitatively different vacuum configurations may be a global minimum of the one-loop effective potential in a large domain of the parameter space.

This concludes theSO proof of existence of all ofM theU group-theH oretically allowed vacua. Nonsupersymmetric models broken at by the SM preserving VEVs, do exhibit at the quantum level the full spectrum of intermediate symmetries. This is (10) 45 crucially relevant for those chains that, allowed by gauge uniﬁcation, are accidentally excluded by the tree level potential. Chapter 3. The quantum vacuum of the minimal SO GUT 112 (10) Chapter 4

SUSY-SO breaking with small representations (10) 4.1 What do neutrinos tell us?

In Chapter 3 we showed that quantum effects solve the long-standingSO issue of the incompatibility between theH dynamicsH of the simplest nonsupersymmetric Higgs sector spanning over ⊕ and gauge coupling unification. (10) In order to give mass to the SM fermions at the renormalizable level one has to H 45 16 H H H minimally add a . So it wouldSO be natural to consider the Higgs sector ⊕ ⊕ as a candidate for the minimal theory, as advocated long ago by Witten [67]. 10 10 16 45 However the experimental data accumulated since the tell us that such an Higgs (10) sector cannot work. It is anyway interesting to review the general idea, especially as 1980 far as concerns the generation of neutrino masses. H First of all with just one the Yukawa lagrangian is 10 Y Y F F H ; L h.c. (4.1) 10 VCKM = 16Y 16 10 + which readily implies , since can be always diagonalized by a rotation in F the flavor space of the . However this10 is not a big issue. It would be enough to add H =1 H a second or even better a which can break the down-quark/charged-lepton 16 symmetry (cf. Eqs. (1.180)–(1.183)). 10 120 The most interesting part is about neutrinos. In order to give a Majorana mass to B L B L H∗ the RH neutrinos − must be broken by two units. Since − h i − this H∗ H∗ F F d means that we have to couple the bilinear to . Such a operator 16 = 1 can be generated radiatively due to the exchange of GUT states [67]. H∗ H∗ 16 16 16 16H∗ = 5 Effectively the bilinear can be viewed as a . So we are looking for F F H∗ states which can connect the matter bilinear with an effective . Since 16 16 126 H V V V ⊗ ⊗ ⊃ a possibility is given by the combination (where are 16 16 126 10 45 45 126 10 45 45 45 Chapter 4. SUSY-SO breaking with small representations 114 (10) SO H V the gauge bosons). IndeedY the and thegU ’s can be respectively attached to the matter bilinear via Yukawa ( ) and gauge ( ) interactions; and to the bilinear H∗ H∗ (10) λ10 45 via scalar potential couplings10 ( ) and again gauge interactions. The topology of the diagram is such that this happens for the first time at the two-loop level (see 16 16 e.g. Fig. 4.1).

Figure 4.1: Two-loop diagram responsible for neutrino masses in the Witten mechanism. Figure taken from [117].

Notice that the same diagram generates also a Majorana mass term for LH neutrinos, while a Dirac mass term arises from the Yukawa lagrangian in Eq. (4.1). At the leading order the contribution to the RH Majorana, Dirac and LH Majorana neutrino MR MD ML mass matrices (respectively , and ) is estimated to be

αU χR u αU χL MR Y λ ; MD Y v ; ML Y λ ; ∼ π 2 M2U ∼ ∼ π 2 M2U (4.2) 10 10 10 10 χR B L H SU vu where is − breaking VEV of the in the singlet direction, ; ; χL ; ; 10 and ∗ are instead electroweak VEVs. After diagonalizing 1 1 16 (5) = the full 2 ×10 neutrino mass matrix2 16 (1 2 + ) = (1 2 + ) M M 6 6 L D ; T MD MR ! (4.3)

ν;νc deﬁned on the symmetric basis , we get the usual type-II and type-I contributions to the × light neutrinos mass matrix ( ) T mν ML MDMR− MD : 3 3 − (4.4) 1 = 4.1. What do neutrinos tell us? 115

ML Y − The type-II seesaw is clearly too small ( ∼ eV) while the type-I is naturally 6 too big 10 u 1 T αU − MU10v MDMR− MD Y λ− Y : ∼ π 2 χR 2 ∼ eV (4.5) 1 1 10 6 10 10 (2 ) u λ αU /π − χL v 10 MU For the estimates we have taken ∼ , ∼ , ∼ ∼ GeV, ∼ GeV χR χR MU 2 2 15 and ∼ GeV, where ≪ by unification constraints.10 13 1 10 10 10 The only chance in order to keep neutrino masses below eV is either to push χR MU10 Y − ∼ or to take ∼ . The first option is unlikely in nonsupersymmet- SO 6 1 ric because of unification10 constraints , while the second one is forbidden by Mu MD 10 2 H the fact that , in the simplest case with just the in the Yukawa sector (10) (cf. e.g. Eqs. (1.180)–(1.183)) . = 10 H H As we have already anticipated the reducible representation ⊕ is needed in the Yukawa sector in order to generate non trivial mixing and break the down- 10 120 quark/charged-lepton symmetry.Mu M InterestinglyD this system would also allow for a disentanglement between and (cf. e.g. Eqs. (1.180)–(1.183)) and a fine-tuning in order toH suppressH neutrinoH massesH is in principle conceivable. However the Higgs sector ⊕ ⊕ ⊕ starts to deviate from minimality and maybe there is a better option to be considered. 10 16 45 120 H H The issue can be somewhat alleviated by considering a in place of a in the Higgs sector, since in such a case the neutrino masses is generated at the F H∗ 126 χR/MU 16 renormalizable level by the term . This lifts the problematic suppression d 2 MR χR MB L factor inherent to the effective mass and yields ∼ ∼ − , that might 16 126 be, at least in principle, acceptable. This scenario, though conceptually simple, involves = 5 a detailed one-loop analysis of the scalar potential governing the dynamics of the H H H ⊕ ⊕ Higgs sector and is subject of an ongoing investigation [74]. We will briefly mention some preliminary results in the Outlook of the thesis. 10 45 126 On the other hand it would be also nice to have a viable HiggsSO sector with only representations up to the adjoint. This is not possible in ordinary , but what about the supersymmetric case? Invoking TeV-scale supersymmetry (SUSY), the qualitative (10) picture changes dramatically. Indeed, the gauge running within the MSSM prefers MB L MU d − in the proximity of and, hence, the Planck-suppressed RH neutrino F H /MP H H mass operator , available whenever ⊕ is present in the Higgs sector, 2 2 MR = 5 can naturally reproduce the desired range for . 16 16 16 16 Accidentally gravity would be responsible for a contribution of the same order of magnitude. Indeed 1 SO d if we take the Plank scale as the cut-off of the theory we find a effective operators F F H∗ H∗ /MP MR χR/MP of the type , which leads to ∼ . The analogy comes from the fact that αU /π − MU MP (10)2 = 5 ∼ . 2 16 16 16 16χR MU In supersymmetry ∼ , but then the two-loop diagram in Fig. 4.1 would disappear due to the ( 2 ) non-renormalization theorems of the superpotential. This brought the authors of Refs. [117, 118] to reconsider the Witten mechanism in the context of split-supersymmetry [197, 198, 199]. Chapter 4. SUSY-SO breaking with small representations 116 (10) SOThis well known fact motivates us to re-examin the issue of the breaking of SUSY- in the presence of small representations. (10) 4.2 SUSY alignment: a case for flipped SO

In the presence of supersymmetry one would naively say that the minimal Higgs sector SO H H H ⊕ ⊕ (10) that sufficesH to break H to the SM is given by . Let us recall that both as well as are required in order to retain SUSY below the GUT scale. (10) 45 16 16 However, it is well known [60, 61, 62] that the relevant superpotential does not 16 16 SO support, at the renormalizable level, a supersymmetric breaking of the gauge group to the SM. This is due to the constraints on the vacuum manifold imposed F D (10) SUby the - and H-flatnessH conditions which, apart from linking the magnitudes of the -singlet and vacuum expectation values (VEVs), make the adjoint VEV H H H SU SO h i aligned to . As a consequence, an subgroup of the initial (5) 16 16 gauge symmetry remains unbroken. In this respect, a renormalizable Higgs sector with H H H H SU 45 ⊕ 16 16 ⊕ (5) (10) H in place of suffersSU from the same “ lock” [62], because also in the SM singlet direction is -invariant. 126 126 16 16 (5) This issue can be addressed by giving up renormalizability [61, 62]. However, this op- 126 (5) tion may be rather problematic since it introduces a delicate interplay between physics MU MP ≪ at two different scales, , with the consequenceMU of splitting the GUT-scale thresholds over several orders of magnitude around . ThisB mayL affect proton decay as well as the SUSY gauge unification, and may force the − scale below the H H d ⊕ GUT scale. The latter is harmful for the setting withH H relying on a RH neutrino mass operator. The models with ⊕ are also prone to trouble 16 16 = 5 with gauge unification, due to the number of large Higgs multiplets spread around the 126 126 GUT-scale. SO Thus, in none of the cases above the simplest conceivable Higgs sector spanned over the lowest-dimensionality irreducible representations (up to the adjoint) (10) seems to offer a natural scenario for realistic model building. Since the option of a simple GUT-scale Higgs dynamics involving small representations governed by a simple renormalizable superpotential is particularly attSUractive, we aimed at studying the conditions under which the seeminglySO ubiquitous lock can be overcome, while keeping only spinorial and adjoint representations. (5) Let us emphasize that the assumption that the gauge symmetry breaking is driven (10) by the renormalizable part of the Higgs superpotential does not clash with the fact that, H H in models with ⊕ , the neutrino masses are generated at the non-renormalizable level, and other fermions may be sensitive to physics beyond the GUT scale. As far 16 16 d as symmetry breaking is concerned, Planck induced ≥ effective interactions are irrelevant perturbations in this picture. SU 5 H H H The simplest attempt to breaking the lock by doubling either ⊕ or (5) 16 16 45 4.2. SUSY alignment: a case for flipped SO 117

F (10) in order to relax the -flatness constraints is easily shown not to work. In theH formerH ⊕ case, there isF only one SM singlet field directionH associated to each of the pairs. Thus, -flatness makes the VEVs in align along this direction regardless H H F ∂W/∂ 16H 16 of the number of ⊕ ’s contributing to the relevant -term, (see for 45 H instance Eq. (6) in ref. [62]). Doubling the number of ’s does not help either. Since 16 16 F 45 H there is no mixing among the 45’s besides the mass term, -flatness aligns both h i SU H H 45 in the direction of ⊕ . For three (and more) adjoints a mixing term of 45 the form is allowed, but it turns out to be irrelevant to the minimization so (5) 16 16 that the alignment1 2 3 is maintained. 45 45 45 From this brief excursus one might conclude that, as far as the HiggsSO content is considered, the price for tractability and predictivity is high on SUSY models, as the desired group-theoretical simplicity of the Higgs sector, with representations up (10) to the adjoint, appears not viable. In this chapter, we pointSO out that all these issues are alleviated if one considers a flippedSO variant ofU the SUSY unification. In particular, we shall show that the flipped ⊗ scenario [71, 72, 73] offers an attractive option to break the gauge (10) symmetry to the SM at theSO renormalizable level; by; means of a quite simpleH Higgs (10) (1) ⊕ sector, namely a couple ofSO spinorsU and one adjoint . Within the extended ⊗ gauge1 2 algebra1 2 one finds in general three in- (10) 16 16 45 equivalent embeddings of the SM hypercharge. In addition to the two solutions with (10) (1)SU SU U SO the hypercharge stretching over the or the SU ⊗ subgroups of (respectively dubbed as the “standard” and “flipped” embeddings), there is a third, SO SO(5) U (5) (1) (10) “flipped” , solution inherent to the ⊗ case, with a non-trivial projection U (5) of the SM hypercharge onto the factor. (10) (10) (1) SU Whilst the difference between the standard and the flipped embedding is se- SO (1) SO mantical from the point of view, the flipped case is qualitatively different. SO (5) In particular, the symmetry-breaking “power” of the spinor and adjoint repre- (10) SO (10) sentations is boosted with respect to the standard case, increasing the number (10) of SM singlet fields that may acquire non vanishing VEVs. Technically, flipping allows H H (10) for a pair of SM singletsH in each of the and “Weyl” spinors, together with four SM singlets within . This is at the root of the possibility of implementing the gauge 16 16 symmetry breaking by means of a simple renormalizable Higgs sector. Let us just 45 remark that, if renormalizability is not required, theSO breaking can be realized without the adjointU Higgs field, see for instance the flipped model with an additional anomalous of Ref. [200]. (10) SU Nevertheless, flipping is not per-se sufficient to cure the lock of standard SO (1)H H H with ⊕ ⊕ in the Higgs sector. Indeed, the adjoint does not reduce (5) the rank and the bi-spinor, in spite of the two qualitatively different SM singlets involved, (10) 16 16 45 SU U can lower it only by a single unit, leaving a residual ⊗ symmetry (the two H SU SM singlet directions in the still retain an algebra as a little group). Only H H H (5) (1) when two pairs of ⊕ (interacting via ) are introduced the two pairs of SM 16 (5) 16 16 45 Chapter 4. SUSY-SO breaking with small representations 118 (10) singlet VEVs in the spinor multiplets may not generally be aligned and the little group is reduced to the SM. Thus, the simplest renormalizableSO SUSY Higgs model that can provide the spontaneous breaking of the GUT symmetry to theSO SM by meansU of Higgs representations not larger than the adjoint, is the flipped ⊗ scenario with (10) two copies of the ⊕ bi-spinor supplemented by the adjoint . Notice further (10) (1) that in the flipped embedding the spinor representations include also weak doublets 16 16 45 that may trigger the electroweak symmetry breaking and allow for renormalizable Yukawa interactions with the chiral matter fields distributed in the flipped embedding over ⊕ ⊕ . Remarkably, the basics of the mechanism we advocate can be embedded in an 16 10 1 E H H underlying non-renormalizable Higgs model featuring a pair of ⊕ and the H adjoint . 6 27 27 Technical similarities apart, there is, however, a crucial difference between the SO 78U E E ⊗ and SOscenarios,U that is related to the fact that the Lie-algebra of is larger than that of 6 ⊗ . It has been shown long ago [201] that the renor-6 (10) (1) E H H H malizable SUSY Higgs model spanned on a single copy of ⊕ ⊕ leaves SO (10) (1) H H an symmetry6 unbroken. Two pairs of ⊕ are needed to reduce the 27 27 78 H rank by two units. In spite of the fact that the two SM singlet directions in the (10) H 27 27 are exactly those of the “flipped” , the little group of the SM singlet directions H H H H H SU 27 ⊕ ⊕ ⊕ and h i remains at the renormalizable level , as 1 1 2 2 16 we will explicitly show. 27 27 27 27 78 (5) HAdding non-renormalizable adjoint interactions allows for a disentanglementE of the h i, such that the little group is reduced to the SM. Since a one-step breaking is phenomenologically problematic as mentioned earlier, we argue for a two-step6 break- 78 SO U E ing, via flipped ⊗ , with the scale near the Planck scale.SO U In summary, we make the case for an6 anomaly free flipped ⊗ partial (10) (1) unification scenario. We provide a detailed discussion of the symmetry breaking pat- SO (10) (1) tern obtainedE within the minimal flipped SUSY Higgs model and consider its possible embedding. We finally present an elementary discussion of the flavour (10) structure offered6 by these settings.

4.3 The GUT-scale little hierarchy

SO H H H In supersymmetric models with just ⊕ ⊕ governing the GUT breaking, one way to obtain the misalignment between the adjoint and the spinors is that of (10) 45 16 16 invoking new physics at the Planck scale, parametrized inM aP model-independent way by a tower of eﬀective operators suppressed by powers of .

What we callMU /M theP “GUT-scale little hierarchy" is the hierarchy induced in the GUT spectrum by suppressed effective operators, which may split the GUT-scale 4.3. The GUT-scale little hierarchy 119 thresholds over several orders of magnitude. In turn this may be highly problematic for proton stability and the gauge unification in low energy SUSY scenarios (as discussed for instance in Ref. [202]). It may also jeopardize the neutrino mass generation in the seesaw scheme. We briefly review the relevant issues here. 4.3.1 GUT-scale thresholds and proton decay

In Ref. [203] the emphasis is set on a class of neutrino-mass-related operators which turns out to be particularly dangerous for proton stability in scenarios with a non- renormalizable GUT-breaking sector. The relevant interactions can be schematically written as

WY F g F H H F f F H H ⊃ MP MP 1 1 vR 16 16 16 16 + 16 16 16 16 QgL T QfQT ; ⊃ MP (4.6) g f vR H +H T T where and are matrices in the family space, ≡|h i| | | and ( ) is the H H color triplet (anti-triplet) contained in the ( ). Integrating out the color triplets, MT 16 = 16 whose mass term is labelled , one obtains the following eﬀective superpotential SU L 16 16 involving ﬁelds belonging to doublets

L vR T T T Weff (2) u Fd′ u GV ′ℓ d′ GV ′ν′ ; MPM2 T − (4.7) 2 u ℓ = where and denote the physical left-handed up quarks and charged lepton superfieldsd′ ν′ in the basis in which neutral gaugino interactions are flavor diagonal. Thed andν fields are related to the physical down quark and light neutrino fields and by d′ VCKM d ν′ VPMNS ν V ′ Vu†Vℓ Vu Vℓ and . In turn , where and diagonalize the left- handed up quark and chargedT lepton mass matrices respectively. The × matrices G;= F =G; F Vu g; f Vu = are given by . g f 3 3 By exploiting the correlations between the and matrices and the matter masses ( ) ( ) = ( ) and mixings and by taking into account the uncertainties related to the low-energy SUSY spectrum, the GUT-thresholds and the hadronic matrix elements, the authors of Ref. [203] argue that the effective operators in Eq. (4.7) lead to a proton lifetime − νK : ; ∼ − × yrs (4.8) 1 + 33 : at the verge of the currentΓ experimental( ) (0 6 lower3) bound10 of × years [80]. In 33 obtaining Eq. (4.8) the authors assume that the color triplet masses cluster about the MT H H MU 0 67 10 GUT scale, ≈ h i∼hSO i ≡ . OnSU the other hand, in scenariosSU where at the renormalizable level is broken to and the residual symmetry 16 45 (10) (5) (5) Chapter 4. SUSY-SO breaking with small representations 120 (10) isSU broken to SM by means of non-renormalizable operators,H /MP the effectiveH /MP scale of the h i h i MU breaking physics is typicallySU suppressed by or with respect to . As a consequence, the -part of the colored triplet higgsino spectrum is (5) MU /MP 16 45 effectively pulled down to the scale, in a clash with proton stability. 2 (5) 4.3.2 GUT-scale thresholds and one-step unification SU The “delayed” residual breakdown has obvious implications for the shape of the gauge coupling unification pattern. Indeed, the gauge bosons associated to the SU /SM (5) coset, together with theMU relevant/MP part− of the Higgs spectrumSO, tend/SU to be uniformly shifted [61] by a factor ∼ below the scale of the (5) 2 MU gauge spectrum, that sets the unification scale, . These thresholds may jeopar- 10 (10) (5) dize the successful one-step gauge unification pattern favoured by the TeV-scale SUSY extension of the SM (MSSM). 4.3.3 GUT-scale thresholds and neutrino masses

With a non-trivial interplay among several GUT-scale thresholds [61] one may in principle end up with a viable gauge unification pattern. Namely, the threshold effects in different SM gauge sectorsMU /MP may be such that unification is preserved at a larger scale. In such a case the suppression is at least partially undone. This,d in turn, is unwelcome for the neutrino mass scale because the VEVs entering the effective F H /MP operator responsible for the RH neutrino Majorana mass term are raised MR MU /MP 2 2 MR=5. accordingly and thus ∼ tends to overshoot the upper limit GeV 2 16 16 14 implied by the light neutrino masses generated by the seesaw mechanism. 10 Thus, although the Planck-induced operators can provide a key to overcoming the SU SO SU C SU L U Y Ï ⊗ ⊗ H lockH of theH minimal SUSY Higgs model with ⊕ ⊕ , such an effective scenario is prone to failure when addressing the (5) (10) (3) (2) (1) measured proton stability and light neutrino phenomenology. 16 16 45 4.4 Minimal flipped SO Higgs model

SO As already anticipated in the previous sections, in a standard framework with a Higgs sector built off the lowest-dimensional(10) representations (up to the adjoint), it is (10) rather difficult to achieve a phenomenologically viable symmetry breaking patternH even admitting multiple copies of each type of multiplets. Firstly, with a singleSU at play, at the renormalizable-level the little group of all SM singlet VEVs is regardless H H 45 of the number of ⊕ pairs. The reason is that one can not get anything more SU SU (5) than an singlet out of a number of singlets. The same is true with a H 16 16 second added into the Higgs sector because there is no renormalizable mixing (5) (5) 45 4.4. Minimal flipped SO Higgs model 121 (10) H among the two ’s apart from the mass term that, without loss of generality, can be taken diagonal. With a third adjoint Higgs representation at play a cubic 45 interaction is allowed. However, due to the total antisymmetry of the invariant1 2 and3 45 45 45 to the fact that the adjoints commute on the SM vacuum, the cubic termSO doesU not contribute to the F-term equations [204]. This makes the simple flipped ⊗ model proposed in this work a framework worth of consideration. For the sake of (10) (1) completeness, let us alsoSO recall that admitting Higgs representations larger than the adjoint a renormalizable → SM breaking can be devised with the Higgs sector H H H H H H H H of the form ⊕ ⊕ ⊕ [205], or ⊕ ⊕ ⊕ [62] for a (10) renormalizable seesaw. 54 45 16 16 54 45 126 126 In Tables 4.1SO and 4.2 we collect a list of theE supersymmetric vacua that are obtained in the basic Higgs models and their embeddings by considering a set of Higgs representations of the dimension of the adjoint6 and smaller, with all SM singlet (10) VEVs turned on. The cases of a renormalizable (R) or non-renormalizable (NR) Higgs potential are compared. We quote reference papers where results relevant for the present study were obtained without any aim of exhausting the available literature. The results without reference are either verified by us or follow by comparison with other cases and rank counting. The main results of this study are shown in boldface.

SO SO U Standard Flipped ⊗ Higgssuperﬁelds R NR R NR SO SU(10) SO U(10) SU (1) U ⊕ ⊗ ⊗ SO SU SO U 16× 16 ⊕ (10) (5) (10) ⊗ (1) SM(5) (1) SU SU U U ⊕ ⊕ [60] SM [61] ⊗ SM ⊗ 2 16 16 SU(10) (5) SM(10) (1) 45 ⊕ 16× 16 ⊕ (5) SM (5) (1) SM (1) 45 2 16 16 (5) Table 4.1: Comparative summary of supersymmetric vacua left invariant by theSO SM singlet VEVs inSO variousU combinations of spinorial and adjoint Higgs representations of standard and ﬂipped ⊗ . The results for a renormalizable (R) and a non-renormalizable (NR) Higgs(10) superpotential are(10) respectively(1) listed.

SOWe are goingU to show thatSO by considering a non-standard hypercharge embedding in ⊗ (flipped ) the breaking to the SM is achievable at the renor- H H H malizable level with ⊕ × ⊕ Higgs fields. Let us stress that what we (10) (1) (10) require is that the GUT symmetry breaking is driven by the renormalizable part of the 45 2 16 16 superpotential, while Planck suppressed interactions may be relevant for the fermion mass spectrum, in particular for the neutrino sector. Chapter 4. SUSY-SO breaking with small representations 122 (10) Higgssuperfields R NR E SO ⊕ E SU × ⊕ 6 27 27 SO (10)U 6 ⊕ ⊕ [201] SM ⊗ 2 27 27 SU 5 (5) 78 ⊕ 27× 27 ⊕ (10) SM (1) E Table 4.2: Same as in78 Table 4.12 for27 the 27gauge group( ) with fundamental and adjoint Higgs representations. 6 4.4.1 Introducing the model Hypercharge embeddings in SO U ⊗ SO (10) (1) UTheX so called flipped realization of the gauge symmetry requires an additional gauge factor in order to provide an extra degree of freedom for the SM hyper- (10) SU C SU L charge identification. For a fixed embedding of the ⊗ subgroup within SO(1) , the SM hypercharge can be generally spanned over the three remaining Car- U SO (3) U (2)X/ SU C SU L tans generating the abelian subgroup of the ⊗ ⊗ (10) 3 coset. There are two consistent implementations of the SM hypercharge within the SO (1) (10) SU(1) ( (3) (2) ) algebra (commonly denoted by standardU X and flipped ), while a third one becomes available due to the presence of . (10) (5) In order to discuss the different embeddings we find useful to consider two bases U (1) for the subgroup. Adopting the traditional left-right (LR) basis corresponding to SU C 3 SU L SU R U B L SO the ⊗ ⊗ ⊗ − subalgebra of , one can span the SM (1) U R U B L U X hypercharge on the generators of ⊗ − ⊗ : (3) (2) (2) (1) (10) Y αT β B L γX: (1)R (1)− (1) (3) (4.9) TR B L The normalization of the and= − +charges( is) + chosen so that the decompositions (3) SO SU C SU L ⊗ ⊗ ofU theR spinorialU B L and vector representations of with respect to ⊗ − read (10) (3) (2) ; ; ; ; ; ; ; ; ; ; (1) (1) ⊕ − ⊕ − − ⊕ − ⊕ ; ; 1 ; 1 1 1 1 1 ⊕ − 3 2 3 2 3 2 16 = (3; 2;0;1+ ) (3; 1; +; ) (;3 1; ; ) ; (1 2;0; ; 1) (1 1; + +1) 2− ⊕ ⊕ ⊕ − (4.10) (1 1; +1)2 2 1 1 3 B3 L T2 2 10 = (3 1;0 ) (3 1;0 + −) (1 2; +R 0) (1 2; 0) which account for the standard SU andU (3)Zassignments. SO ⊗ U Alternatively,Y U Z U consideringX the SO U subalgebraX of , we identify the ′ ⊗ ⊗ subgroup of ⊗ , and equivalently write: (5) (1) (10) Y αY ′ βZ γX ; (1) (1) (1) (10) (1) (4.11) = ˜ + ˜ + ˜ 4.4. Minimal flipped SO Higgs model 123 (10) Y ′ Z SU C SU L U Y U Z where and are normalized so that the ⊗ ⊗ ′ ⊗ analogue of eqs. (4.10) reads: (3) (2) (1) (1) ; ; ; ; ; ; ; ; ; ; ⊕ − ⊕ − ⊕ − − ⊕ ; ; 1 ; 1 2 1 ⊕ 6 3 3 2 16 = (3; 2; + ; +1) (3; 1; + ; 3) (3; 1; ; +1) (1; 2; ; 3) : (1 1;+1 +1) − − ⊕ ⊕ − ⊕ − (4.12) (1 1;0 +5)1 1 1 1 3 U 3 2 2 X 10 = (3 1; 2) X(3 1; + +2) (1 2; + 2) (1 2; +2) In both cases, the charge hasX been convenientlyX fixed to SO for the spinorial representation (and thus − and also for the16 vector (1) = +1 U X and singlet, respectively; this is also the10 minimal way to ob1 tain an anomaly-free , SO U X = 2 =E +4 (10) that allows ⊗ to be naturally embedded into ). (1) It is a straightforward exercise to show that inc order to acco6 mmodatec the SM quark (10) (1) Q ; ; u ; ; d ; ; multiplets with quantum numbers , − and there 1 2 1 are only three solutions. 6 3 3 U = (3 2 + ) = (3 1 ) = (3 1 + ) On the bases of Eq. (4.9) (and Eq. (4.11), respectively) one obtains, 3 (1) α ;β ; γ ; α ; β ; γ ; (4.13) 1 =1 = 2 =0 ˜ =1 ˜ =0 ˜ =0 SO which is nothing but the “standard” embedding of the SM matter into . Explicitly, Y TR B L Y Y ′ SU − in the LR basis (while in the picture). (3) 1 (10) The second2 option is characterized by = + ( ) = (5) α ;β ; γ ; α ; β ; γ ; − − (4.14) 1 1 1 = 1 = 2 =0SU ˜ = 5 ˜ = 5 ˜ =0 which is usually denoted “flipped ” [69, 70] embedding because the SM hyper- SU U Z SO Y Z Y ′ charge is spanned non-trivially on the ⊗ subgroup of , − . SU C SU L(5) SU R U B L 3 1 Remarkably, from the ⊗ ⊗ ⊗ − perspective this setting5 cor- SU R (5) (1) TR Y (10) TR = ( B L) responds to a sign flip of the Cartan operator , namely − − (3)π (2) SU(2) R (1) (3) (3) 1 which can be viewed as a rotation in the algebra. 2 (2) = + ( ) A third solution corresponds to (2) α ;β ; γ ; α ; β ; γ ; − − − (4.15) 1 1 1 1 1 =0SO = 4 = 4 ˜ = 5 ˜ = 20 ˜ = 4 denoted as “flipped ” [71, 72, 73] embedding of the SM hypercharge.γ γ Notice, in particular, the fundamental difference between the setting (4.15) with and (10) U X 1 the two previous cases (4.13) and (4.14) where does not play any role. 4 Y = ˜ = Analogously to what is found for , once we consider the additional anomaly-free U X (1) gauge factor, there are three SM-compatible ways of embedding the physical G (1)By definition, a flipped variant of a specific GUT model based on a simple gauge group is obtained 3 G U by embedding the SM hypercharge nontrivially into the ⊗ tensor product. (1) Chapter 4. SUSY-SO breaking with small representations 124 (10) B L SO U X SU − into ⊗ . Using the compatible description they are respectively given by (see Ref. [206] for a complete set of relations) ( ) (10) (1) (5) B L Y ′ Z ; − (4.16) B L 1 Y ′ Z X ; − 5 − (4.17) (B L) = 1 (4 Y+′ ) Z X : − −20 − − (4.18) ( ) = 1(16 +5 ) 20 B L ( ) = (8 3 5 ) where the first assignmentY B isL the standard − embedding in Eq. (4.9). Out of × possible pairs of and − charges only do correspond to the quantum numbers SO 3 3 of the SM matter [206]. By focussing on the flipped hypercharge embedding ( ) B L 6 in Eq. (4.15), the two SM-compatible − assignments are those in Eqs. (4.17)–(4.18) TR (10) B L (they are related by a sign flip in ). In what follows we shall employ the − ((3) ) assignment in Eq. (4.18). ( ) Spinor and adjoint SM singlets in flipped SO

U X B L The active role of theSO generator in the SM(10) hypercharge (and − ) identification within the flipped scenario has relevant consequences for model building. In (1) SO particular, the SM decomposition of the representations change so that there (10) H H H are additional SM singlets both in ⊕ as well as in . (10) SO The pattern of SM singlet components in flipped has a simple and intuitive SO U16 X 16E 45 interpretation from the ⊗ ⊂ perspective, where ⊕ − (with the U X (10) subscript indicating the charge) are contained6 in ⊕ while+1 is1 a part of E (10) (1) 16 16 the adjoint . The point is that the flipped SM hypercharge assignment0 makes (1) E 27 27 45 the various6 SM singlets within the complete representations “migrate” among their SO 78 E different sub-multiplets; namely, theSO two6 SM singlets in the of that in the standard embedding (4.13) reside in the singlet and spinorial components6 (10) SO 27 both happen to fall into just the single ⊂ in the flipped case. (10) 1 16 SO Similarly, there are two additional SM singlet directions in in the flipped SO 16 27 (10) scenario, that, in the standard embedding, belong to the 0− ⊕ components E 45 (10) of the of , thus accounting for a total of four adjoint SM singlets.3 +3 (10) 16 16 In Tables 4.3,6 4.4 and 4.5 we summarize the decomposition of the − , and 78 SO U X representations of ⊗ under the SM subgroup, in both the2 standard+1 SO SU 10 16 and0 the flipped cases (and in both the LR and descriptions). The pattern 45 (10) (1) of the SM singlet components is emphasized in boldface. (10) (5) The supersymmetric flipped SO model

TheSU presence of additional SM singlets(10) (some of them transforming non-triviallySO under ) in the lowest-dimensional representations of the flipped realisation of the gauge symmetry provides the ground for obtaining a viable symmetry breaking with (5) (10) 4.4. Minimal flipped SO Higgs model 125 (10) SU LR SO SO f SO SO f ; ; ; (5); − − − − ; (10)1 ; (10)1 ; (10)1 ; (10)1 3 6 − 3 6 3 5 − 3 5 (3; 1; 1 ) (3; 1; 2) (3; 1; 1) (3; 1; 1 ) 3 6 − 3 6 2 5 − 2 5 (3; 1; + 1 ) + (3; 1; 1 ) + (1; 2; + 1) (1; 2; 2 ) − 2 1− − 2 1− − 3 5 − 3 5 (1 2; + 1 ) (1 2; 1 ) (3 1; + 1) (3 1; 1 ) 1 1 5 5 2 2 2 2 SU C SU L Table 4.3: Decomposition(1 2; of the) fundamental(1 2; 10-dimensional) (1 2; ) representatio(1 2; n) under ⊗ ⊗ U Y SO SO U X SO f , for standard and flipped ⊗ ( ) respectively. In the first two columns SU C (3) (2) ± (LR)(1) the subscripts keep(10) track of the (10)origin(1) of the multiplets(10) (the extra symbols correspond to TR SU the eigenvalues of the Cartan generator) while in the last two columns the content is shown. (3) (4) SU (5) LR SO SO f SO SO f ; ; ; (5); ; (10)1 ; (10)1 ; (10)1 ; (10)1 − 6 4 6 4 − 3 5 3 5 (3; 2; + 1) (3; 2; + 1 ) (3; 1; + 1 ) (3; 1; + 1 ) 2 4 2 4 2 5 2 5 (1; 2; 1) + (1; 2; + 1 ) + (1; 2; 1 ) (1; 2; + 1 ) 4− 4− 10 10 (3 1;− + 3) (3 1; + 3 ) (3 2; +− 6 ) (3 2; + 6 ) ; 2 1; 1 0 1 ; 2 1; 1 0 101 3 4 43 4 3 10 3 10 (3 1; ) + (3 1; + +) (3 1; ) (3 1; + ) 1; 1 0 1; 1 0 1; 1 0 1 1; 1 0 1 4−4 4− 10 (1 1; +1) ( ; ) (1 1; +1) ( ; ) ( ; ) ( ; ) ( ; ) ( ; ) Table 4.4: The same as in Table 4.3 for the spinor SU-dimensional representation.e ; The SMν singlets; are emphasized in boldface and shall be denoted, in the description, as ≡ and ≡ . e ν 16 SU R 10 1 The LR decomposition shows that and belong to an(5) doublet. (1 1; 0) (1 1; 0) (2) a significantly simplified renormalizable Higgs sector. Naively, one may guess that the H H D pair of VEVs in (plus another conjugated pair in to maintain the required - flatness) might be enough to break the GUT symmetry entirely, since one component 16 SU SO 16 SU transforms as a of ⊂ , while the other one is identified with the singlet (cf. Table 4.4). Notice that even in the presence of an additional four-dimensional 10 (5) (10) (5) vacuum manifold of the adjoint Higgs multiplet, the little group is determined by the H F VEVs since, due toH the simple form of theH renormalizableH superpotential -flatness makes the VEVs of align with those of , providing just enough freedom for 16 them to develop non-zero values. 45 16 16 Unfortunately, this is still not enough to support the desired symmetry breaking H SU U pattern. The two VEV directions in are equivalent to one and a residual ⊗ H SO U symmetry is always preserved by h i [195]. Thus, even in the flipped ⊗ 16 (5) (1) 16 (10) (1) Chapter 4. SUSY-SO breaking with small representations 126 (10) SU LR SO SO f SO SO f (5) 1; 1 0 10 1; 1 0 10 1; 1 0 1 1; 1 0 1 (10) (10) (10) (10) 1; 1 0 15 1; 1 0 15 1; 1 0 24 1; 1 0 24 ( ; ; ) ( ; ; ) ( ; ; ) ( ; ; ) ( ; ; ) ( ; ; ) ( ; ; ) ( ; ; ) 15 −15 −24 24 (8; 1;0)2 (8; 1;0)1 (8; 1;0)5 (8; 1;0)1 − 3 15 3 15 6 24 − 6 24 (3; 1; + 2) (3; 1; 1 ) (3; 2; 5 ) (3; 2; + 1 ) 3 15 3 15 6 24 6 24 (3; 1; ) (3; 1; + ) (3; 2; + ) (3; 2; ) 1 1 24 24 (1; 3;0)1 + (1; 3;0)1 + (1; 3;0)1 (1; 3;0)1 6 6 − 6 6 − 6 10 6 10 (3 2; + ) + (3 2; + ) + (3 2; + ) (3 2; + ) ; 5 1; 1 0 1 ; 2 1; 1 0 101 6 6 6 6 3 10 3 10 (3; 2; + ) + (3; 2; +) (3; 1; ) (3; 1; + ) − 1− − − − 10 − (1; 1; +1)1 ( ; ; )1 (1; 1; +1)1 ( ; ; )1 − 6 6− 6 6− 6 10 − 6 10 (3; 2; 5) (13; 12; 0 1 ) (3; 2; 2 ) (31; 2;1 0 1 ) 6− 1−6 10 1010 (3 2; − 6) (3 2; + 6 ) (3 1; +− 3 ) (3 1; 3 ) 1 10 Table 4.5: The same(1 as in1; Table1) 4.3( for the; ) representation.(1 1; 1) The( SM; singlets) are given in boldface ωB L ; ω ; ωR ; ω− ; − and labeled throughout the text as − ≡ , ≡ , ≡ and ≡ 45 + + 0 ω ωR ω− where again the LR notation has been used. The15 LR decomposition1 also shows1 that , and 1 SU R ωB L (1B 1;L 0) (1 1; 0) (1 1; 0) + (1 1; 0) belong to an triplet, while − is a − singlet. (2)

H H H SU U setting the Higgs model spanned on ⊕ ⊕SO suffers from an ⊗ lock analogous to the one of the standard SUSY models with the same Higgs 16 16 45 (5) (1) sector. This can be understood by taking into account the freedom in choosing the SO (10) basis in the algebra so that the pair of VEVs within can be “rotated” onto a single component, which can be then viewed as the direction of the singlet in the (10) SU 16 SOdecomposition of ⊕ ⊕ with respect to an subgroup of the original gauge symmetry. 16 = 5 10 1 H H(5) ⊕ (10)On theH other hand, with a pair of interacting ’s the vacuum directions in the two ’s need not be aligned and the intersection of the two different invariant SU 16 16 subalgebras (e.g. , standard and flipped for a specific VEV configuration) leaves as 16 SU C SU L U Y F a little group the ⊗ ⊗ of the SM. -flatness makes then the adjoint H H (5) VEVs ( is the needed carrier of interaction at the renormalizable level) aligned (3) (2) (1) H H H to the SM vacuum. Hence, as we will show in the next section, × ⊕ 45 16 SO U X defines the minimal renormalizable Higgs setting for the SUSY flipped ⊗ 2 (16 + 16 )SO45 model. For comparison, let us reiterate that in the standard renormalizable SU (10) H (1)H setting the SUSY vacuum is always regardless of how many copies of ⊕ (10) (5) 16 16 4.4. Minimal flipped SO Higgs model 127 (10) are employed together with at most a pair of adjoints. The matter sector

Due to the flipped hypercharge assignment,SO the SM matter can no longer be fully embedded into the 16-dimensional spinor, as in the standard case. By inspecting Table 4.4 one can seec thatc in the flipped setting thec pair of the SM sub-multiplets of u e (10) d transforming as and is traded for an extra -like state and an extra SM singlet. SO 16 The former pair is instead found in the vector and the singlet (the lepton doublet as well appears in the vector multiplet).SO Thus, flipping spreads each of the SM ⊕ ⊕ (10) matter generations across of , which, by construction,E SO can be viewedU X as the complete 27-dimensional fundamental representation of ⊃ ⊗ . 16 10 1 (10) ; ; ; ; This brings in a set of additional degrees of freedom, in particular 6 , , ; ; ; ; ; ; SO(10) (1)1 , − and − , where the subscript indicates their 16 origin.3 16 1 1 1 (1 1 0) (3 1 + ) Notice,2 however,16 that3 10 these SM “exotics”2 10 can be grouped into superheavy vector-like (1 2 + ) (3 1 ) (1 2 ) (10) UpairsX and thus no extra states appear in the lowSO energyU spectrX um. Furthermore, the anomalies associated with each of the ⊗ matter multiplets cancel when summed over the entire reducible representation ⊕ − ⊕ . An elementary (1) (10) (1) discussion of the matter spectrum in this scenario is deferr1 ed to2 Sect.4 4.6. 16 10 1 4.4.2 Supersymmetric vacuum

The most general renormalizable Higgs superpotential, made of the representations ⊕ ⊕ ⊕ ⊕ is givenµ by 1 1 2 W2H ρij i j τij i j ; 45 16 16 16 16 Tr (4.19) i;j ; 2 where and the notation= is45 explained+ 16 in16 Appendix+ 16 45 F.1.16 Without loss of general- µ 2 τ ρ ity we can take real by a global phase redefinition, while (or ) can be diagonalized =1 2 by a bi-unitary transformation acting on the flavor indices of the and the . Let us τij τiδij τi choose, for instance, , with real. We label the SM-singlets contained in the e ; SO 16 ν ;16 ’s in the following way: ≡ (only for flipped ) and ≡ (for = all embeddings). 10 1 16 (1 1;0) ωR ωB L ω ω− e ; (10)e ; ν ; (1ν ;1;0) By plugging in the SM-singlet VEVs , − , , , , , and (cf. Ap- + pendix F.1), the superpotential on the vacuum reads 1 2 1 2 1 2 1 2 WH µ ωR ωB L ω−ω h i − ρ e 2e ν 2ν ρ e +e ν ν ρ e e ν ν ρ e e ν ν = 2 +3 +4 ωR ωB L τ11 1ω−1e ν 1 1 ω ν 21e 2 1 e2 e1 ν 12ν 1 2 −1 2 e e 22 ν 2ν 2 2 2 + (− + − ) + ( − √+ )−+ ( +√ ) + ( + ) + 1 1 1 1 1 ωR 1 1 1 1 3 ωB L 1 1 1 1 + τ ω−e ν ω ν e (e e ν ν ) + − (e e + ν ν ) : − − − √2 − 2 √2 (4.20) + 2 2 2 2 2 2 2 2 2 3 2 2 2 2 + ( ) + ( + ) 2 2 2 Chapter 4. SUSY-SO breaking with small representations 128 (10) In order to retain SUSY down to the TeV scale we must require that the GUT gauge symmetryD breakingF preserves supersymmetry. In Appendix F.2 we work out the relevant -ρ and τ-term equations. We find that the existence of a nontrivial vacuum requires (andD for consistency) to be hermitianSO matrices. This is a consequi encei of∗ the fact that -term flatness for the flipped embedding implies h i (see Eq. (F.30) and the discussion next to it). With this restriction the vacuum manifold (10) 16 = 16 is given by i φe φν i φe φν µω τ r α e − τ r α e − ;

i φ1e φ1ν i2 φe 2 φν µω−+ τ r2 α e−( − ) τ r2 α e( − −) ; 1 1 1 2 2 2 8 = sin2 1 1+ sin2 2 2 √ µω τ r2 α (τ r ) α ;2 ( ) R 1 1 1 2 2 2 8 = sin2 + sin2 √ 2 2 µωB L 1τ r τ r1 ; 2 2 − − 1 − 2 4 2 = cos2 +iφ cos2 e r 2 α e2 e ; ; ; ; 1 1 ;2 2 1 2 4 2 = iφν ; ν1;2 r1;2 α 1; 2 e ; 1 2 = cos iφe ; e1;2 r1;2 α1 2; e− ; 1 2 = sin iφν ; ν1;2 r1;2 α 1; 2 e− ; = cos 1 2 (4.21) 1 2 1 2 1 2 r ; α± α α where and ≡ ±= aresin ﬁxed in terms of the superpotential parameters, 1 2 1 2 µ ρ τ ρ τ r − ; − τ τ (4.22) 2 22 1 11 2 1 2µ (ρ τ 2 5ρ τ ) r = 1−2 ; − 3τ τ (4.23) 2 11 2 22 1 2 2 ( ν25 ) e =α− ξ 1 2− ; 3 ν e − (4.24) sinΦν sinΦe cos α = ξ ; sin (Φν Φe) (4.25) + − sinΦ + sinΦ cos = with sinρ (Φ Φ ) ξ | | : ρ τ ρ τ 12 ρ ρ (4.26) τ2 τ2 − 2 −6 1 = q 5 11 5 22 ν e 1 2 22 11 The phase factors and are deﬁned as + 26

ν φν φν φρ ; e φe φe φρ ; Φ ≡ Φ − ≡ − (4.27) 1 2 12 1 2 12 φν ; φe ; φρ in terms of the relevantΦ phases + , andΦ . Eqs.+ (4.24)–(4.25) imply that for ν e α− ξ α , Eq. (4.24) reduces to 1 2 1 2 Ï 12 while is undetermined (thus + parametrizing an orbit of isomorphic vacua). Φ = Φ = Φ cos cosΦ 4.4. Minimal flipped SO Higgs model 129 (10)

In order to determine the little group of the vacuum manifold we explicitly computeα− 6 the correspondingν e gauge boson spectrum in Appendix F.3. We find that, for and/or 6 , the vacuum in Eq. (4.21) does preserve the SM algebra. = 0 As already mentioned in the introduction this result is a consequence of the mis- Φ =Φ alignement of the spinorH VEVs, that is made possible at the renormalizable levelα by− the interaction with the . If we choose to align the ⊕ and ⊕ VEVs ( ν e and ) or equivalently, to decouple one of the1 Higgs1 spinors2 from2 the vacuum r 45 SU U 16 16 16 16 =0 ( for instance) the little group is ⊗ . Φ = Φ 2 ThisH resultH can be easily understoodH by observingH that in the case with just one pair =0 ⊕ ⊕(5) (1) eofH νH (or with two pairsSU of R aligned) the two SM-singlet directions, and , are connected by an transformation. This freedom can be used to 16 16 16 16 SU U rotate one of the VEVs to zero, so that the little group is standard or flipped ⊗ , (2) depending on which of the two VEVs is zero. (5) (1) In this respect, the Higgs adjoint plays the role of a renormalizable agentSU thatU prevents the two pairs of spinor vacua from aligning with eachF other along the ⊗ direction. Actually, by decoupling the adjoint Higgs, -flatness makes the (aligned) i i F ⊕ (5) (1) vacuumH trivial, as one verifiesρ by inspecting the -terms in Eq. (F.14) of Ap- pendix F.2 for h i and det 6 . 16 16 H H The same result with just two pairs of ⊕ Higgs multiplets is obtained by 45 =0 =0 adding non-renormalizable spinor interactions, at the cost of introducing a potentially 16 16 SO critical GUT-scale threshold hierarchy. In the flipped setup here proposed the GUT symmetry breaking is driven by the renormalizable part of the Higgs superpoten- (10) tial, thus allowing naturally for a one-step matching with the minimal supersymmetric extension of the SM (MSSM). E Before addressing the possible embedding of the model in a unified scenario, we comment in brief on the naturalness of the doublet-triplet mass splitting6 in flipped embeddings. 4.4.3 Doublet-Triplet splitting in flipped models

Flipped embeddings offers a rather economical way to implement the Doublet-Triplet (DT) splitting through the so called Missing Partner (MP) mechanism [207, 208]. In SU U Z ⊗ order to show the relevat features let us consider first the flippedSU U Z . In order to implement the MP mechanism in the flipped ⊗ the Higgs (5) (1) superpotential is required to have the couplings WH ; (5) (1) ⊃ − − − (4.28) +1 +1U 2 Z 1 1 +2 where the subscripts correspond10 to10 the 5 + 10quantum10 5 numbers, but not the − mass term. From Eq. (4.28) we extract the relevant terms that lead to a mass for2 the+2 (1) 5 5 Higgs triplets WH ; ; ; ; ; ; : ⊃h i − h i − (4.29) 1 1 1 1 10 10 5 10 10 5 (1 1;0) (3 1; + 3 ) (3 1; 3 ) + (1 1;0) (3 1; 3 ) (3 1; + 3 ) Chapter 4. SUSY-SO breaking with small representations 130 (10) On the other hand, the Higgs doublets, contained in the − ⊕ remain massless since they have no partner in the ⊕ − to couple with.2 +2 5 SO5 The MP mechanism cannot be+1 implemented1 in standard . The relevant 10 10 SO interactions, analogue of Eq. (4.28), are contained into the invariant term (10) WH ; ⊃ (10) (4.30) which, however, gives a mass to the161610+ doublets as16 well,1610 via the superpotential terms WH ; ; ; ; ; ; : ⊃h i − − (4.31) 1 1 1 1 116 16 510 116 516 10 SO U X 2 5 2 2 2 5 Flipped(1 1;0) ⊗(1 2; ,) on(1 the2;+ other) + hand,(1 1;0) offers again(1 2; + the) possibility(1 2; ) of implementing the MP mechanism. The prize to pay is the necessity of avoiding a large (10) (1) number of terms, both bilinear and trilinear, in the Higgs superpotential. In particular, the analogue of Eq. (4.28) is given by the non-renormalizable term [200]

WH : ⊃ MP MP (4.32) 1 1 2 2 1 1 1 2 2 1 16 16 16 16 + 16 16 16 16 By requiring that ( ) takes a VEV in the ( ) direction while ( ) in the ( ) component,1 one1 gets 16 16 2 2 16 16 1 1 16 16 16 16 10 10 WH ; ⊃ MP h i MP h i (4.33) 2 2 1 1 1 161 16 16 161 1 16 162 162 16 1 10 10 5 + 1 10 10 5 which closely resembles Eq. (4.28), leading to massive triplets and massless doublets. In order to have minimally one pair of electroweak doublets, one must further require that the × mass matrix of the ’s has rank equal to one. Due to the active role of non-renormalizable operators, the Higgs triplets turn out to be two orders of magnitude 2 2 SO U X 16 below the flipped ⊗ scale, reintroducing the issues discussed asSO in Sect. 4.3. An alternative possibility for naturally implementing the DT splitting in is the (10) (1) Dimopoulos-Wilczek (DW) (or the missing VEV) mechanism [209]. In order to explain SO (10) the key featuresSU itC is convenientSU L SU to decomposeR the relevant representations in terms of the ⊗ ⊗ group (10) ; ; ; ; ::: (4) (2) ≡(2) ⊕ ⊕ ; ; ; ; ; ≡ ⊕ 45 (1; 1; 3) (15; ;1 1); ≡ ⊕ 16 (4; 2; 1) (4; 1; 2) ; ≡ ⊕ 16 (4 2 1) (4 1 2) (4.34) ωR ; ; ωB L ; ; SO where ≡h i and − 10≡h (6 1 1)i. In(1 the2 2) standard case (see [210, 211] SU L and [212] for a recent discussion) one assumes that the doublets are contained in (1 1 3) (15 1 1) (10) two vector multiplets ( and ). From the decompositions in Eq. (4.34) it’s easy to see (2) 1 2 10 10 4.5. Minimal E embedding 131 6 that the interaction SU L (where the antisymmetry of ωR requires the presence of two ’s) leaves the 1 doublets2 massless provided that . For the naturalness 10 4510 45 of the setting other superpotential terms must not appear, as a direct mass term for 10 (2) =0 one ofSU the ’s and theω interactionR ωB L term . The latter aligns the SUSY vacuum in the direction ( − ), thus destabilizing the DW solution. 10 1645 16 On the other hand, the absence of the interaction enlarges the global (5) = symmetries of the scalar potential with the consequent appearance of a set of light 1645 16 pseudo-Goldstone bosons in the spectrum. To avoid that the adjoint and the spinor sector must be coupled in an indirect way by adding extra fields and symmetries (see for instance [210,SO 211, 212]).U X Our flipped ⊗ setting offers the rather economical possibility of em- H bedding the electroweak doublets directly into the spinors without the need of (10) (1) (see Sect. 4.6). As a matter of fact, there exists a variant of the DW mechanism where SU L H H 10 the doublets, contained in the ⊕ , are kept massless by the condition ωB L F − (see e.g. [213]). However, in order to satisfy in a natural way the -flatness for (2) ωB L 16 16 the configuration − , again a contrived superpotential is required, when com- =0 pared to that in Eq. (4.19). In conclusion, we cannot implement in our simple setup = 0 any of the natural mechanisms so far proposed and we have to resort to the standard minimal fine-tuning. 4.5 Minimal E embedding

SO U 6 ⊗ TheE natural andH minimal unified embeddingH H of the flipped model is with one and two pairs of ⊕ in the Higgs sector. The three matter F (10) (1) families6 are contained in three chiral superfields. The decomposition of the 78 27 27 and representations under the SM quantum numbers is detailed in Tables 4.6 and 27 27 4.7, according to the different hypercharge embeddings. 78 SO In analogy withe the flipped; ν discussion,; we shall label the SM-singlets contained in the as ≡ and ≡ . (10) 11 116 ⊕ ⊕ ⊕ ⊕ SU As we are27 going to show,(1 1;0) the little group(1 1;0 of) E is SUSY- in the renormalizable case. This is just a consequence 1 of2 the larger1 2 algebra. 78 27 27 27 27 In order to obtain a SM vacuum, we need to resort to a non-renormalizable6 scenario (5) H thatSO allowsU for a disentanglement of the h i directions, and, consistently,E for a flipped ⊗ intermediate stage. We shall make the case for an gauge symmetry 78 SO broken near the Planck scale, leaving an effective flipped scenario6 down to the (10) (1) GeV. 16 (10) 10 Chapter 4. SUSY-SO breaking with small representations 132 (10)

SU SU f SO f ; ; ; (5) (5)− (10) ; 1 16 ; 2 16 ; 1 16 (3 1; +− 3 )5 (3 1; − 3 )5 (3 1; + 3 )5 ; 1 16 ; 1 16 ; 1 16 (1 2; 2 )5 (1 2; 2 )5 (1 2; + 2 )5 ; 1 ; 1 ; 1 1016 1016 1016 (3 2;− + 6 ) (3 2; + 6 ) (3 2; + 6 ) ; 2 16 ; 1 16 ; 1 16 3 10 3 10 3 10 (3; 1; ) (3; 1; + ) (3; 1; + ) 1016 1016 1016 (1 1; +1) (1 1; 0) (1 1; 0) ; ; ; 116 116 116 (1 1;− 0) (1 1; +1)− (1 1;− 0) ; 1 ; 1 ; 1 510 510 510 (3 1; 3 ) (3 1; − 3 ) (3 1; − 3 ) ; 1 ; 1 ; 1 510 510 510 (1 2; + 2 ) (1 2; 2 ) (1 2; − 2 ) ; 1 10 ; 1 10 ; 2 10 (3 1; +− 3 )5 (3 1; + 3 )5 (3 1; − 3 )5 ; 1 10 ; 1 10 ; 1 10 (1 2; 2 )5 (1 2; + 2 )5 (1 2; 2 )5 11 11 11 E SU C SU L U Y (1 1; 0) (1 1; 0) (1 1; +1) ⊗ ⊗ Table 4.6: Decomposition of the fundamental representation of under f , 6 according to the three SM-compatible diﬀerentSU embeddingsSO of27 the hypercharge(3) ( stands(2) for ﬂipped).(1) The numerical subscripts keep track of the and origin. (5) (10) 4.5. Minimal E embedding 133 6

SU SU f SO f ; ; ; (5) (5) (10) ; ; ; 11 11 11 (1 1; 0) (1 1; 0) (1 1; 0) ; ; ; 145 145 145 (1; 1; 0) (1; 1; 0) (1; 1; 0) 2445 2445 2445 (8 1; 0)− (8 1; 0) (8 1; 0) ; 5 ; 1 ; 1 2445 2445 2445 (3 2; 6 ) (3 2;− + 6 ) (3 2;− + 6 ) ; 5 45 ; 1 45 ; 1 45 6 24 6 24 6 24 (3; 2; + ) (3; 2; ) (3; 2; ) 2445 2445 2445 (1; 3; 0) (1; 3; 0) (1; 3; 0) 2445 2445 2445 (1 1; 0) (1 1; 0)− (1 1; 0) ; 1 ; 5 ; 1 1045 1045 1045 (3 2;− + 6 ) (3 2; − 6 ) (3 2; + 6 ) ; 2 45 ; 2 45 ; 1 45 3 10 − 3 10 3 10 (3; 1; ) (3; 1; ) (3; 1; + ) 1045 1045 1045 (1 1;− +1) (1 1; 1) (1 1;− 0) ; 1 45 ; 5 45 ; 1 45 (3 2; 6 )10 (3 2; + 6 )10 (3 2; − 6 )10 ; 2 45 ; 2 45 ; 1 45 (3 1; +− 3 )10 (3 1; + 3 )10 (3 1; 3 )10 ; 45 ; 45 ; 45 (1 1; 1)10 (1 1;− +1)10 (1 1;− 0)10 ; 1 16 ; 2 16 ; 2 16 (3 1; +− 3 )5 (3 1; − 3 )5 (3 1; − 3 )5 ; 1 16 ; 1 16 ; 1 16 (1 2; 2 )5 (1 2; 2 )5 (1 2; − 2 )5 ; 1 ; 1 ; 5 1016 1016 1016 (3 2;− + 6 ) (3 2; + 6 ) (3 2; − 6 ) ; 2 16 ; 1 16 ; 2 16 3 10 3 10 − 3 10 (3; 1; ) (3; 1; + ) (3; 1; ) 1016 1016 1016 (1 1; +1) (1 1; 0) (1 1; −1) ; ; ; 116 116 116 (1 1; 0)− (1 1; +1) (1 1; 1) ; 1 ; 2 ; 2 516 516 516 (3 1; 3 ) (3 1; + 3 ) (3 1; + 3 ) ; 1 ; 1 ; 1 516 516 516 (1 2;− + 2 ) (1 2;− + 2 ) (1 2; + 2 ) ; 1 ; 1 ; 5 16 16 16 (3 2; 6 )10 (3 2; − 6 )10 (3 2; + 6 )10 ; 2 ; 1 ; 2 − 3 1016 3 1016 3 1016 (3; 1; + ) (3; 1; ) (3; 1; + ) 16 16 16 (1 1; 1)10 (1 1; 0)−10 (1 1; +1)10 116 116 116 (1 1; 0) (1 1; 1) (1 1; +1) Table 4.7: The same as in Table 4.6 for the representation. 78 Chapter 4. SUSY-SO breaking with small representations 134 4.5.1 Y and B L into E (10) − E 6 Interpreting the differentSU possibleC SU definitionsL SU R of the SM hypercharge in terms of the maximal subalgebra ⊗ ⊗ , one finds that the three assignments in6 SU I Eqs. (4.13)–(4.15) are each orthogonal to the three possible ways of embedding I R; R′;E SU(3) R (3) (3) (with ) into [206]. Working in the Gell-Mann basis (cf. Appendix G.1) SU R (2) the Cartan generators read = (3) ′ ′ (3) TR T T ; ′ − ′ (4.35) (3) 1 1 2 ′ ′ ′ TR 2 1T 2T T ; = √ ′ ′ − ′ (4.36) (8) 1 1 2 3 2 3 1 2 3 SU R = SU+ R′ 2 SU E which defines the embedding. The ′ and ′ embeddings′ ′ are obtained from Eqs. (4.35)–(4.36) by flipping respectively ↔ and ↔ . Considering the SO(2) (2) (2) standard and flipped embeddings of the hypercharge in Eq. (4.13) and Eq. (4.15), SU 2 3 3 1 in the notation they are respectively given by 3 (10) Y T T T T T ; (3) √ L R √ R √ L √ E − (4.37) 1 (8) (3) 1 (8) 1 (8) 2 (8) = 3 + + 3 = 3 3 and Y T T T T T : √ L − √ R √ L E √ E (8) (8) (8) (3) (8) (4.38) 1 2 1 B L1 = 3 3 = 3 + +− 3 Analogously, the three SM-compatible assignments of SU inI Eqs. (4.16)–(4.18)SU R are as well orthogonal to the three possible ways of embedding into . However, BonceL we fix the embedding of the hypercharge we haveY only twoB coL nsistent choices for − (2)− (3) available.SU TheyI correspond to the pairs where and are not orthogonal to the same [206]. B L For the standard hypercharge embedding, the − assignment in Eq. (4.16) reads (2) B L TL TR TL TE TE ; − √ √ − − √ (4.39) 2 (8) (8) 2 (8) (3) 1 (8) B L = 3 + = 3 3 SO while the − assignment in Eq. (4.18), consistent with the flipped embedding of the hypercharge, reads (10) B L TL TR TR TL TE : − √ − − √ √ (4.40) 2 (8) (3) 1 (8) 2 (8) (8) 3 3 3 4.5.2 The E vacuum= manifold = +

The most general6 renormalizable Higgs superpotential, made of the representations ⊕ ⊕ ⊕ ⊕ , is given by 1 µ2 1 2 78 27WH 27 27 ρ27ij i j τij i j αijk i j k βijk i j k ; Tr (4.41) 2 = 78 + 27 27 + 27 7827 + 27 27 27 + 27 27 27 2 4.5. Minimal E embedding 135 6 i;j ; αijk βijk ijk where . The couplings and are totally symmetric in , so that each one of them contains four complex parameters. Without loss of generality we can µ = 1 2 τ take real by a phase redeﬁnition of the superpotential, while can be diagonalized

τbyij aτ bi-unitaryiδij τ transformationi actingα onβ the indices of the and the . We take, , with real. Notice that and are not relevant for the present study, since 27 27 the corresponding invariants vanish on the SM orbit. = In the standard hypercharge embedding of Eq. (4.37), the SM-preserving vacuum directions are parametrized by a a b a T ′ a T ′ T ′ T ′ T ′ T ′ T ′ T T T ; h i ′ ′ √ ′ ′ − ′ √ ′ − ′ √ − (4.42) 3 2 3 1 2 3 4 1 2 3 1 2 3 1 2 2 3 1 2 3 1 2 1 2 3 78 = + + ( + 2 ) + ( ) + ( + 2 ) and 6 2 6 i ei v νi v ; h i ′ ′ ′ ′ (4.43) i ei u3 νi u3 : 3 2 (4.44) 27 = ( ) 3 + ( ) 2 a a a a b e e ν ν 3 3 ; ; 27 ; = ( ; ) + ( ) where , , , , , B , L , and are 13 SM-singlet VEVs (see Appendix G.1 for notation).1 2 Given3 4 the3 1 2− 1 2expression1 2 1 in2 Eq. (4.39) and the fact that we can rewrite the Cartan part of h i as √ a TR 78 √ a b TR TL √ a b TR TL ; − − (4.45) (3) 1 (8) (8) 1 (8) (8) 4 3 3 3 3 2 + 2 ( + )SO + + 2 ( ) we readilyE identify theωR standarda ωB L a VEVsb used in thea previousb sectionSO withU theX present notation as ∝ , − ∝ , while ∝ − is the ⊗ E TX TR TL (10) singlet VEV6 in ( ∝ −4 ). 3 3 3 3 (8) (8) + Ω (10) (1) We can also write6 SO the vacuum manifold in such a way that it is manifestly invariant under′ ′ the flipped hypercharge in Eq. (4.38). This can be obtained by flipping ↔ in Eqs. (4.42)–(4.44), yielding (10) 1 3 a T ′ a T ′ √ a′ T a′ b T T h i ′ ′ E √ E L 1 2 (3) 1 (8) (8) 1 2 3 2 1 4 2 3 a′ b TE TL ; 78 = + + 2 + ( + ) +√ − − (4.46) 1 (8) (8) 3 3 + 2 ( ) i ei v νi v ; h i ′ ′ (4.47) 3 ′ 3 ′ i ei u νi u ; 1 2 (4.48) 27 = ( ) 1 + ( ) 2 B L 3 3 27 = ( ) + ( ) where we recognize the − generator defined in Eq. (4.40). Notice thatSO the Cartan subalgebraY is actually invariant both under the standard and the flipped form of . We have a′ TE a′ TE a TR a TR ; (10) (8) (3) (8) (3) (4.49) 3 4 3 4 + = + Chapter 4. SUSY-SO breaking with small representations 136 (10) with a′ a √ a ; − − (4.50) a √ a a 3′ 3 4 2 = − 3 (4.51) a a ; ′ ; 4 3 4 thus making the use of or 2directions= 3 in the+ flipped or standard vacuum manifold 3 4 3 4 Ecompletely equivalent. WeSO can now completeU X the identification of the notationω± useda ; for with that ofE the flipped ⊗ model studied in Sect. 4.4, by ∝ . 6 From the stand point, the analyses of the standard and flipped vacuum manifolds1 2 (10) (1) given, respectively,6 in Eqs. (4.42)–(4.44) and Eqs. (4.46)–(4.48), lead, as expected, to the same results with the roles of standard and flipped hypercharge interchanged (see Appendix G). In order to determine the vacuum little group we may therefore proceed with the explicit discussion of the standard setting. By writing the superpotential in Eq. (4.41) on the SM-preserving vacuum in Eqs. (4.42)– (4.44), we find a a b WH µ a a h i 2 2 2 (4.52) 3 4 3 ρ e e 1ν ν2 ρ e e ν ν ρ e e ν ν ρ e e ν ν = + + + 2 2 2 a ν ν τ11 1a 1e ν 1 1a ν e21 2 1 a 2 e1 e 12 ν1ν2 1 2 22 2 2b e2e2 ν ν + (− + − ) + ( + ) + − ( + √) + −( + ) " r 4 1 1 r # 1 1 1 1 2 1 1 2 3 1 1 1 1 1 2 3 1 1 1 1 + + + a ν ν ( + ) τ a e ν a ν e 3a e e 2ν ν 2 3b e e ν ν : − − − √ − " r 4 2 2 r # 2 1 2 2 2 2 2 2 3 2 2 1 2 2 2 3 2 2 2 2 + + D +F ( + ) When applyingρ theτ constraints3 coming from2 - andSO-term2 equations,3 a nontrivial vacuum exists if and are hermitian, as in the flipped case. This is a consequence D i i ∗ of the fact that -flatness implies h i (see Appendix G.2 for details). D F (10) E After imposing all the constraints due to - and -flatness, the vacuum manifold 27 = 27 can be finally written as 6 i φν φe i φν φe µa τ r α e − τ r α e − ;

i φ1ν φ1e i2 φν 2 φe µa τ r2 α e−( − ) τ r2 α e( − −) ; 1 1 1 1 2 2 2 2 = sin2 1 1+ sin2 2 2 √ µa τ 2r α ( ) τ r 2 α ( ; ) 2 1 1 1 2 2 2 2 = − sin2 −+ sin2 √ µa τ r2 α τ r α 2; 3 1 1 1 2 2 2 2 6 = − (3 cos 2− + 1) (3 cos 2 + 1) √ µb √ τ 2r 2√ τ r ; 2 2 4 1 1 1 2 2 2 2 = sin iφ sin e r 2α e e ; 2; ;3 ; 1 1 ; 2 2 1 2 3 = 2 + 2iφν ; ν1;2 r1;2 α 1; 2 e ; 1 2 = cos iφe ; e1;2 r1;2 α1 2; e− ; 1 2 = sin iφν ; ν1;2 r1;2 α 1; 2 e− ; = cos 1 2 (4.53) 1 2 1 2 1 2 = sin 4.5. Minimal E embedding 137 6 r ; α± α α where and ≡ ± are ﬁxed in terms of superpotential parameters, as follows 1 2 1 2 µ ρ τ ρ τ r − ; − τ τ (4.54) 2 22 1 11 2 1 µ(ρ τ 2 4ρ τ ) r = −1 2 ; − 5τ τ (4.55) 2 11 2 22 1 2 ( 4ν2 ) e =α− ξ 1 2 − ; 5 ν e − (4.56) sinΦν sinΦe cos α = ξ ; sin (Φν Φe) (4.57) + − sinΦ + sinΦ cos = with sinρ (Φ Φ ) ξ | | : ρ τ ρ τ 12 ρ ρ (4.58) τ2 τ2 − 2 −5 1 = q 4 11 4 22 ν e 1 2 22 11 The phase factors and are deﬁned as + 17

ν φν φν φρ ; e φe φe φρ : Φ ≡ Φ − ≡ − (4.59) 1 2 12 1 2 12 Φ + Φ + In AppendixSU G.3 we show that the little group of the the vacuum manifold in Eq. (4.53) is . H It is instructive to look at the configuration in which one pair of , let us say (5) τ ρ ρ ⊕ , is decoupled. This case can be obtained by setting α27 in2 the relevant2 equations. In agreement with Ref. [201], we find that2 12turns22 out to 27 27 F = = = 0 be undetermined by the -term constraints, thusα parametrizingα a set of1 isomorphic solutions. We may therefore takeSO in Eq. (4.53) and show that the little group corresponds in this case to (see Appendix1 G.3),2 thus recovering the result = = 0 of Ref. [201]. (10) The same result is obtained in the case in which the vacua of the two copies of H H α− ν e ⊕ αare aligned, i.e. and . Analogously to the discussion in Sect. 4.4.2, is in this case undetermined and it can be set to zero, that leads us again 27 27 +H H =SO 0 Φ = Φ to the one ⊕ case, with as the preserved algebra. These results are intuitively understood by considering that in case there is just 27 H 27 H (10) i i one pair of ⊕ (or the vacua of the two pairs of ⊕ are aligned) the e ν SU R SM-singlet directions and are connected by an transformation which can 27 27 27 27 SO be used to rotate one of the VEVs to zero, so that the little group is locked to an H (2) H e ν configuration. On the other hand, two misaligned ⊕ VEVs in the − plane SU (10) lead (just by inspection of the VEV quantum numbers) to an little group. SO 27 27 In analogy with the flipped case, the Higgs adjoint plays the role of a renor- i i (5) malizable agent that prevents the two pairs of ⊕ from aligning within each SO (10) F other along the vacuum. Actually, by decoupling the adjoint Higgs, -flatness 27 27 (10) Chapter 4. SUSY-SO breaking with small representations 138 (10) i i F ⊕ makes the (aligned) vacuumH trivial, as oneρ verifies by inspecting the -terms in Eq. (G.18) of Appendix G.2 for h i and det 6 . 27 27 E SU In conclusion, due to the larger algebra, the vacuum little group remains , 78 =0 =0 never landing to the SM. In this respect6 we guess that the authors of Ref. [214], who H H H (5) advocate a ⊕ × ⊕ Higgs sector, implicitly refer to a non-renormalizable setting. 78 2 27 27 4.5.3 Breaking the residual SU via effective interactions SU In this section we considerE the possibility(5) of breaking the residual symmetry of the renormalizable vacuum through the inclusion of effective adjoint Higgs inter- MP SO(5) U X actions near the Planck6 scale . We argue that an effective flipped ⊗ ≡ SO f Mf may survive down to the ≈ GeV scale, with thresholds spread in MP Mf 16 (10) (1) between and in such a way not to affect proton stability and lead to realistic (10) 10 neutrino masses. E ME

WH λ λ ::: ; MP Tr Tr (4.60) NR h 2 2 4 i 1 1 2 = 78 + 78 + whereD the ellipses stand for terms which include powers of the ’s representations and ≥ operators. Projecting Eq. (4.60) along the SM-singlet vacuum directions in 27 Eqs. (4.42)–(4.44) we obtain 5

WH λ a a a a b MP NR n 2 2 2 2 λ1 a1a 1a 2a 3a 4a 3 a a a a b ::: : = 2 + + + √ (4.61) h 2 2 2 2 1 1 2 2 2 1 4i o 2 1 2 1 2 3 4 3 4 3 4 3 + 2 + + + 3 + 2 + + 2 F + One verifies that including the non-renormalizable contribution in the -term equations allows for a disentanglement of the h i and ⊕ ⊕ ⊕ VEVs, so that the breaking to the SM is achieved. In particular, the1 SUSY1 vacuu2m allows2 for an interme- SO f 78 27 27 27 27 diate stage (that is prevented by theF simple renormalizable vacuum manifold in Eq. (4.53)). By including Eq. (4.61) in the -term equations, we can consistently neglect (10) SO U all VEVs but the ⊗ singlet , that reads µM (10) (1) Ω P : − λ λ (4.62) 2 1 Ω = 1 2 2 E It is therefore possible to envisage a scenario5 + where the symmetry is broken at ME < MP SO U X a scale leaving an effective flipped ⊗ 6 scenario down to the GeV, as discussed in Sect. 4.4. All remaining SM singlet VEVs are contained in 16 (10) (1) 10 4.5. Minimal E embedding 139 6 ⊕ ⊕ ⊕ ⊕ Mf ME that are the only Higgs multiplets required to survive at the ≪ 1 scale.1 It is2 clear2 that this is a plausibility argument and that a detailed study 45 16E 16 16 16 of the vacuum and related thresholds is needed to ascertain the feasibility of the 6 scenario. E SO f The non-renormalizableMf breaking of through an intermediate stage driven by ≫ , while allowing (as we shall6 discuss next) for a consistent unification (10) pattern, avoids the issues arising within a one-step breaking. As a matter of fact, the Ω D /MP > Mf colored triplets responsible for proton decay live naturally at the 2 scale, while the masses of the SM-singlet neutrino states which enter the "extended" = 5 Mf Ω type-I seesaw formula are governed by the h i ∼ (see the discussion in Sect. 4.6). 4.5.4 A unified E scenario 27

Let us examine the plausibility6 of the two-step gauge unification scenario discussed in the previous subsection. We consider here just a simplified description that neglectsSO f thresholds effects. As a first quantitative estimate of the running effects on the couplings let us introduce the quantity (10) α M α M X− f − f bX b ME Mf − − ; ≡ 1 α− 1 α− π Mf ˆ E 10 E ˆ 10 (4.63) ( ) 1 ( ) 1 1 M E∆( ) α = E log U E E 2 X where is the unification scaleX andX/√is the gauge coupling. The charge has been properly6 normalized to . The6 one-loop beta coefficients for the H H H F F F G (1) superfield content ⊕ × ⊕ ⊕ × ⊕ ⊕ ⊕ are found to be b bX / ˆ = 24 and . 45 2 16 16 3 (16 10 1 ) 45 10 Taking, forˆ the sake of an estimate, a typical MSSM value for the GUT coupling αE− =1 M=E/M 67f 24< Mf < ≈ , for one finds . 1 2 SO f In order to match the couplings with the measured SM couplings, we 25 10 ∆( ) 5% consider as a typical setup the two-loop MSSM gauge running with a 1 TeV SUSY scale. (10) The (one-loop)Mf matching of the non abelian gauge couplings (in dimensional reduction) at the scale reads α− Mf α− Mf α− Mf ; (4.64) 1 1 1 10 2 Y 3 while for the properly normalized( hypercharge) = ( ) = one( obtains)

α− Mf α β α− Mˆf γ α− Mf : Y X (4.65) 1 2 2 1 2 1 ˆ 10 ˆ ( ) = ˆ + ˆ ( ) + ˆ ( ) Here we have implemented the relation among the properly normalized U(1) generators (see Eq. (4.15)) Y αY ′ βZ γX; (4.66) α; β; γ ; ; √ ˆ ˆ ˆ ˆ ˆ with { } {− − }. = ˆ + + ˆ 1 1 q 3 3 ˆ ˆ ˆ = 5 5 2 10 Chapter 4. SUSY-SO breaking with small representations 140

-1 Αi (10)

29 UH1LY 28

UH1LX 26 SUH2LL

25 SOH10L E6

24 SUH3LC log10HΜGeVL 15.0 15.5 16.0 16.5 17.0 17.5 18.0

E Figure 4.2: Sample picture of the gauge coupling unification in the -embedded SO U X ⊗ model. 6 (10) (1) The result of this simple exercise is depicted in Fig. 4.2. Barring detailed threshold effects, it is interesting to see that the qualitative behavior of theSO relevant gaugeU X couplings is, indeed, consistentE with the basic picture of the flipped ⊗ embedded into a genuine GUT emerging below the Planck scale. (10) (1) 6 4.6 Towards a realistic flavor

The aim of this section is to provide an elementary discussion of theSO mainU featuresX and of the possible issues arising in the Yukawa sector of the flipped ⊗ model under consideration. In order to keep the discussion simple we shall consider a basic H H (10) (1) Higgs contents with just one pair of ⊕ . As a complement of the tables given in Sect. 4.4, we summarize the SM-decomposition of the representations relevant to the 16 16 Yukawa sector in Table 4.8. For what follows, we refer to [215, 216, 217, 218] and references thereinE where the basic featuresSO of models with extended matter sector are discussedSO inU the and the standard context.U For a scenario employing flipped ⊗ (with6 an additional anomalous ) see Ref. [200]. (10) (10) (1) (1) 4.6.1 Yukawa sector of the flipped SO model

ZConsidering for simplicity just one pair of spinor(10) Higgs multiplets and imposing a matter-parity (negative for matter and positive for Higgs superﬁelds) the Yukawa 2 4.6. Towards a realistic flavor 141

SO SO f c c c c c c c c F D L U Q E N D Q S N ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ (10)c c (10) c F L U ⊕ 5 ⊕ ⊕ 10 1 ⊕ ⊕5 ⊕ 10 1 16 ( ) ( ) ( ) ( c Λ ) (∆ ) ( ) F S E 10 (∆ Λ )5 (∆ Λ)5 (∆ )5 ( Λ)5 H Hd νH Hu sH νH 1 1 h1 i ( )⊕ h i ⊕ ⊕ ⊕ ⊕ ( ⊕) h i ⊕ ⊕ ⊕ ⊕ H Hu νH Hd sH νH 1 1 16 (0 ⊕ h i)5 ⊕ (0 ⊕ 0 ⊕ 0)10 ⊕ ( ) (0 ⊕ h i)5 ⊕ (0 ⊕ 0 ⊕ )10 ⊕ ( ) 1 1 16 (0 )5 (0 0SO 0)10 ( ) (0 )5 (0 0 )10 ( ) SM decomposition of representations relevant for the Yukawa sector in the standard Table 4.8: SO f B L and flipped hypercharge embedding. In the case − is assigned according to Eq. (4.18). A (10) SU SU L self-explanatory SM notation is used, with the outer subscripts labeling thec c origin.c The Q U; D L N; E(10) ; − ; doublets decompose as , , and . Accordingly, Hu ; vu Hd vd; D 0 (5)+ sH 0 νH (2) h i h i and = ( . The) -flatness= ( constraint) Λ = (Λ onΛ the) SM-singletΛ = VEVs,(Λ Λ and) , is taken into account.= (0 ) = ( 0) d superpotential (up to operators) reads WY YU F =5F H YE F F H H YD F F H H ; MP (4.67) 1 = 16 10 16 + 10 1 16 16 + 16 16 16 16 where family indexes are understood. Notice (cf. Table 4.9) that due to the flipped embedding the up-quarks receive mass at the renormalizable level, while all the other fermion masses need Planck-suppressed effective contributions in order to achieve a realistic texture. F F H F F H H F F H H h i c c c c F F H QU S Hu F F H H E Hd νH F F H H N N ν ⊃ h i h i ⊃ h i ⊃ H 16 10 16 c 10 1 16 16 c 16 16 16 16 F F H N L Hu F F H H LE Hd sH F F H H SSs2 ⊃ h i h i⊃ h i ⊃ H (1) 10 5 5 (c + c Λ) (2) 5 1 5 1 Λ (1) 1 1 1 1 c F F H D L νH F F H H SN sH2νH h i⊃ ⊃ (1) 1 5 5 c (2) 5 1 10 5 (1) 10 10 10 10 c c F F H sH F F H H Hd Hd h i⊃ h ih i⊃ h ih i (1) 5 5 1 ( ∆c + Λ ) (4) 10 1 10 1 c c F F H sH F F H H S QD Hd sH h i⊃ h i⊃ h i (1) 5 5 10 Λ Λ (1) 5 5 5 5 Λ Λ c F F H H Q Hd νH h i ⊃ h i (1) 10 5 10 ∆ ∆ (4) 10 5 10 5 c(Λc + ) F F H H N Hd νH (2) 10 10h 5i 1 ⊃ ∆h i (4) 5 1 5 1SU Λ Table 4.9: Decomposition of the invariants in Eq. (4.67) according to flipped and SM. The number F F H H in the round brackets stands for the multiplicity of the invariant. The contractions H H (5) F F 10 1 and yield no SM invariant. 5 1 10 10 16 16 5 1 10 10 Mass matrices

/MP In order to avoid the recursive factors we introduce the following notation for vd vd/MP νH νH /MP sH sH /MP Mf the relevant VEVs (see Table 4.8): ≡ , ≡ and ≡ . The - 1 SU C U Q scale mass matrices for the matter ﬁelds sharing the same unbroken ⊗ ˆ ˆ ˆ (3) (1) Chapter 4. SUSY-SO breaking with small representations 142 (10) quantum numbers can be extracted readily by inspecting the SM decomposition of the relevant matter multiplets in the ﬂipped SO(10) setting: 1+10+16 Mu YU vu ;

YDνH vd YDsH vd Md = ; YU sH YU νH ! ˆ ˆ = YE νH vd YU sH Me ; YE sH vd YU νH ! (4.68) ˆ = YUˆsH YU vu

 YU νH YU vu  0 0 0 Mν  YU sH YU νH YDvdvd YDvdνH YDvdsH  ;   (4.69)  0Y 0v Y ν v Y ν ν Y ν0 s   U u D H d D H H D H H  =  ˆ 2 ˆ 2 ˆ   YU vu YDsH vd YDsH νH YDsH sH   0 2 ˆ ˆ 2 ˆ    c YD YD/ YE YE / U U where, for convenience,c wec redeﬁned0 2 ˆÏ 2 candˆ c Ï ˆ . The basis Mu D; ; D Md −;E E ; Me is used for , for and c c for . The Majorana mass Mν ;N; ;N ; S2 + 2 ( )( ) matrix is written in the basis . ( ∆)(∆ ) 0 (Λ0 )( Λ ) Eﬀective mass matrices (Λ Λ )

Mf sH νH Below the ∼ ∼ scale, the exotic (vector) part of the matter spectrum decouples and one is left with the three standard MSSM families. In what follows, we shall use the calligraphic symbol M for the × eﬀective MSSM fermion mass matrices in order to distinguish them from the mass matrices in Eqs. (4.68)–(4.69). i) Up-type quarks: 3 3 The eﬀective up-quark mass matrix coincides with the mass matrix in Eq. (4.68) u YU vu : M (4.70) ii) Down-type quarks and charged leptons: = The × mass matrices in Eqs. (4.68)– (4.68) can be brought into a convenient form by means of the transformations 6 6 Md MdUd† Md′ ; Me Ue∗Me Me′ ; Ï ≡ Ï ≡ (4.71)

Ud;e Md′ Me′ where are × unitary matrices such that and are block-triangular 6 6 v v v Md′ ; Me′ : O Mf ! O v Mf ! (4.72) 0 = = 0 4.6. Towards a realistic flavor 143

v Here denotes weak scale entries. This corresponds to the change of basis dc c e U ; U − ; c d c e ! ≡ D ! − ! ≡ E ! (4.73) ∆ Λ ∆˜ Λ˜ in the right-handed (RH) down quark and left-handed (LH)d chac rgede lepton sectors, respectively. The upper components of the rotated vectors ( and ) correspond to the light MSSM degrees of freedom. Since the residual rotations acting on the LH Md;e′ down quark and RH charged lepton components, that transformv/Mf the matrices into fully block-diagonal forms, are extremely tiny (of O ), the × upper-left blocks (ULB) in Eq. (4.72) can be identiﬁed with the eﬀective light down-type quark and d Md′ ULB e( M)e′ ULB 3 3 charged lepton mass matrices, i.e., M ≡ and M ≡ . Ud Ue It is instructive to work out the explicit form of the unitary matrices and . ( ) ( ) For the sake of simplicity, in what follows we shall stick to the single family case and assume the reality of all the relevant parameters. Dropping same order Yukawa factors as well, one writes Eqs. (4.68)–(4.68) as

vν vs vν sH Md ; Me ; sH νH ! vs νH ! (4.74) = = Ud Ue and the matrices and are explicitly given by α α Ud;e − : α α ! (4.75) cos sin = Md′ Me′ By applying Eq. (4.71) we get that sinand coshave the form in Eq. (4.72) provided α sH /νH that . In particular, with a speciﬁc choice of the global phase, we can write νH sH tan = α ; α ; sH νH sH νH (4.76)

cos = p 2 v/M2 f sin = p 2 2 so that the mass eigenstates (up to O+ eﬀects) are ﬁnally+ given by (see Eq. (4.73)) c c c d νH sH D ( ) − ; c c c ! sH νH sH νH D ! (4.77) 1 ∆ = p 2 2 ∆˜ + ∆ + and e νH − sH E − ; − ! sH νH sH − νH E ! (4.78) 2 1 2 Λ = p vν;s Λ˜ + OΛ + whereMf the upper (SM) components have mass of and the lower (exotic) ones of O . ( ) ( ) Chapter 4. SUSY-SO breaking with small representations 144

iii) Neutrinos: (10) Mν Working again in the same approximation, the lightest eigenvalue of in Eq. (4.69) is given by νH sH sH νH mν MPvu : ∼ 2 sH νH2 s2H νH2 2 (4.79) 2 ( +2 2 ) 2+2 2 sH νH Mf MP vu For ∼ ∼ ∼ GeV ∼3 (GeV+ and) ∼ GeV one obtains 16 18 2 vu 10 mν 10 : ; 10 ∼ Mf /M2 P ∼ eV (4.80) 2 0 1 which is within the ballpark of the current lower bounds on the light neutrino masses set by the oscillation experiments. ν It is also useful to examine the composition of the lightest neutrinoMν eigenstateν . At the leading order,ν thex light; x ; neutrino x ; x ; x eigenvector obeys the equation which, in the components , reads =0 1 2 3 4sH5x ; = ( ) (4.81) νH x ; 3 (4.82) sH x =0νH x ; 3 (4.83) νH νH x=0 νHsH x ; 1 2 (4.84) sH νH+x sH=0sH x : 4 5 ˆ + 2ˆ =0 (4.85) 4 5 x x 2ˆ + ˆ =0 By inspection,x Eqs. (4.84)–(4.85) are compatible only if , while Eqs. (4.81)– (4.82) imply . Thus, the non-vanishing components4 of the5 neutrino eigenvector x x = = 0 are just and3 . From Eq. (4.83), up to a phase factor, we obtain = 0 νH sH 1 2 ν − N: ν s ν s (4.86) H H 0 H H = p 2 2 Λ ν+ p 2 2 Notice that the lightest neutrino eigenstate+ and the+ lightest charged lepton show the same admixtures of the corresponding electroweak doublet components. Actually, this vu vd SU L can be easily understood by taking theUe limit in which;N the preserved gauge symmetry imposes the same transformation on the components. Ex- Ue = =0 0 (2) plicitly, given the form of in Eq. (4.75), one obtains in the rotated basis (Λ )

 Mf  000M 0 0 Mν′  f  ;  M M  (4.87)  0 0 0f 0 f   MP2 MP2  =  0 0 0M M 0   f f   MP2 MP2   0 0 0 2  00 02 4.6. Towards a realistic flavor 145

c c sH νH Mf Mν′ ν; ; ;N ; S where we have taken ∼ ∼ . is defined on the basis , 0 0 where ν N ˜ − : ( Λ Λ ) ! √ 0 N ! (4.88) 1 Λ 0 = ν0 In conclusion, we see that the "light"Λ˜ eigenstate2 Λ +decouples from the heavy spectrum, c mν M /MP ν N S ; ∼ − f ∼ √ − c (4.89) M 2 1 1 mν 1 Mf /MP νM √ N S ; ∼ · ∼ 2 ( ) (4.90) M 2 2 1 c mν 2 Mf νM √ ; ∼3 − ∼ 2 ( +− ) (4.91) PD 1 1 0 c0 mν 1 Mf νPD √ ; ∼ ∼ 2 (Λ˜ Λ ) (4.92) 1 0 0 PD2 PD2 ν ν 2 (Λ˜ + Λ ) O where and ν are twoν Majorana neutrinos of intermediate mass,O GeV, 1 2 14 while theM states M and form a pseudo-DiracW ν e neutrino of mass of GeV. 1 2 L L L (1016 ) Notice finally thatPD the chargedPD current coupling is unaffected (cf. Eq. (4.86) (10 ) with Eq. (4.78)), contrary to the claim in Refs. [215] and [216], that are based on the ¯ e E unjustified assumption that the physical electron is predominantly made of . Chapter 4. SUSY-SO breaking with small representations 146 (10) Outlook: the quest for the minimal nonsupersymmetric SO theory

H H H In the previous chapters we argued that an Higgs sector(10) basedSOon ⊕ ⊕ has all the ingredients to be the minimal nonsupersymmetric theory. We are 10 45 126 going to conclude this thesis by mentioning some preliminary results of ongoing work (10) and future developments. The first issue to be faced is the minimization of the scalar potential.SO Though there exist detailed studiesH H of the scalarH spectrum of nonsupersymmetric HiggsH sectors based on ⊕ ⊕ [219, 220], such a survey is missing in the ⊕ H H (10) ⊕ case. The reason can be simply attributed the tree level no-go which was 10 54 126 10 plaguing the class of models with just the adjoint governing the first stage of the GUT 45 126 breaking [56, 57, 58, 59]. On the other hand the results obtained in Chapter 3 show thatH the situationH H is drastically changed at the quantum level, making the study of the ⊕ ⊕ scalar potential worth of a detailed investigation. H H ⊕ 10 We45 have126 undertaken such a computation in the case of the scalar potential and some preliminary results are already available [74]. The first technical 45 126 H trouble in such a case has to do with the group-theoretical treatment of the , H especially as far as concerns the invariants. The presence of several invariants 4 126 in the scalar potential is reflected in the fact that there are many SM sub-multiplets H H 126 SO into the ⊕ reducible representation and each one of them feels the breaking in a different way. Indeed the number of real parameters is 16 and apparently, 45 126 H H (10) if compared with the 9 of the ⊕ system (cf. e.g. Eqs. (3.4)–(3.5)), one would think that predictivity is compromised. However, out of these 16 couplings, 3 are fixed by 45 16 the stationary equations, 3 contribute only to the mass of SM-singlet states and 3 do not contribute at all to the scalar masses. Thus we are left with 7 real parameters governing the 22 scalar states that transformH non-triviallH y under the SM gauge group. After imposing the gauge hierarchy h i≫h i, required by gauge unification, the GUT-scale spectrum is controlled just by 4 real parameters while the intermediate-scale 45 126 spectrum is controlled by the remaining 3. Notice also that these couplings are not completely free since they must fulfill the vacuum constraints, like e.g. the positivity of the scalar spectrum. Outlook: the quest for the minimal nonsupersymmetric SO theory 148 (10) H H The message to take home is that in spite of the complexity of the ⊕ system one cannot move the scalar states at will. This can be considered a nice counterex- 45 126 SO ample to the criticism developed in [176] about the futility of high-precision calculations. (10) Actually the knowledge of the scalar spectrum is a crucial information in view of the (two-loop) study of gauge coupling unification. The analysis of the intermediate scales performed in Chapter 2 was based on the ESH [82]: at every stage of the symmetry breaking only those scalars are present that develop a VEV at the current or the subsequent levels of the spontaneous symmetry breaking, while all the other states are clustered at the GUT scale . In this respect the two-loop values obtained for MB L MU αU− 4 − , and in the case of the two phenomenologically allowed breaking chains 1 were

MU MB L − SO C L R B L • ÊÏH − ÊÏH h i h i SM (10) 45 3 2 2 1 126 MB L : ; MU : ; αU− : ; − × GeV × GeV 9 16 1

MU =3 2 MB10L =1 6 10 = 45 5 − SO C L R • ÊÏH ÊÏH h i h i SM (10) 45 4 2 1 126 MB L : ; MU : ; αU− : : − × GeV × GeV 11 14 1 =2 5 10 =2 5 10 = 44 1

MTakenB L atC faceL R B valueL bothC theL R scenarios are in troubleMU eitherC L R because of a too small − ( − and B L case) or a too small ( case). Strictly speaking the lower bound on the − breaking scale depends from the details of the Yukawa 3 2 2 1 4 2 1 MB L & ÷ 4 2 1 sector, but it would be natural to require − GeV. On the other hand the 13 14 d lower bound on the unification scale is sharper since it comes from the gauge 10 MU & : induced proton decay. This constraint yields something like × GeV. 15= 6 Thus in order to restore the agreement with the phenomenology2 one3 10 has to go beyond the ESH and consider thresholds effects, i.e. states which are not exactly clustered at the GUT scale and that can contribute to the running. Let us stress that whenever we pull down a state from the GUT scale the consistence with the vacuum constraints must be checked and it is not obvious a priori that we can do it. C L R B L − For definiteness let us analyze the case. A simple one-loop analyticalMB L survey of the gauge running equations yields the following closed solutions for − 3 2 2 1 With the spectrum at hand one can verify explicitly that this assumption is equivalent to the require- ment4 of the minimal number of fine-tunings to be imposed onto the scalar potential, as advocated in full generality by [83]. Outlook: the quest for the minimal nonsupersymmetric SO theory 149 (10) MU and MB L π α− α− aB L aR α− α− aC α− α− aL − − − − ; M − Z 1 1 2 3221 3 3221 1 1 3221 1 1 3221! 2 2 3 5 + 5 + 1 2 + 3 1 MU = exp π α− α− a α− α− a α− α− a − Y − ∆C − L ; MB L 1 1 SM 1 1 SM 1 1 SM − − ! 2 2 3 + 1 2 + 3 1 = exp (4.93) ∆ with

aL aC aB L aR aY aL aC aC aY aL ; − − − − (4.94) SM SM 3221 3221 SM SM 3221 SM SM 3221 2 3 ∆ = α ; ; + + + MZ where are the5 properly5 normalized gauge couplings at the scale, while aC;L;Y aC;L;R;B L and1 2 3 − are respectively the one-loop beta-functions for the SM and the CSML R B L 3221 − gauge groups. α− α− : α− α− The values of the gauge couplings are such that − ∼ , − ∼ 3 :2 2 α1− α− : 1 1 1 1 , − ∼ − and, assuming the field content 2 of the3 ESH (cf. e.g.1 Table2 2.2) , 1 < 1 21 1 we have 3 1 . Then as long as remains negative when lowering new states below 29 4 50 5 the GUT scale, the fact that the matter fields contribute positively to the beta-functions MB L ∆ 0 − ∆ leadsSU L usSU to concludeC SU thatR U isB L increased (reduced) by the states charged under ( or or − ). MB L C L R B L Thus, in order maximize the raise of − , we must select among the − (2) (3) H (2) H (1) aL > aC;R;B L sub-multiplets of ⊕ those fields with − . The best candidate turns ; ; ; 3221H 3221 3 2 2 1 out to be the scalar multiplet ⊂ . By pulling this color sextet down to MB L 45 126 1 the scale − , we get at one-loop 3 (6 3 1 + ) 126 MB L : ; MU : ; αU− : ; − × GeV × GeV 12 15 1 which is closer to=8 a phenomenologically6 10 reasonable=5 5 10 benchmark. In order= 41 for3 the color sextet to be lowered we have to impose a fine-tuning which goes beyond that needed for the gauge hierarchy. It is anyway remarkable that the vacuum dynamicsMB L allows such a configuration. Another; ; allowed threshold that helps in increasing − is given by the scalar triplet which can be eventually pulled down till to the TeV scale. A full treatment of the threshold patterns is still ongoing [74]. (1 3 0) H What about the addition of a in the scalar potential? Though it brings in many H H new couplings it does not change the bulk of the ⊕ spectrum. The reason is H 10 simply because the can develop only electroweak VEVs which are negligible when 45 126 H H compared with the GUT (intermediate) scale one of the ( ). Thus we expect H 10 H H that adding a will not invalidate the conclusions about the vacuum of the ⊕ 45 126 scalar potential, including the threshold patterns. Of course that will contribute to the 10 45 126 mass matrices of the isospin doublets and color triplets whichd are crucial for other issues like the doublet-triplet splitting and the scalar induced proton decay. =6 Outlook: the quest for the minimal nonsupersymmetric SO theory 150 (10) The other aspect of the theory to be addressed is the Yukawa sector. Such a program has been put forward in Ref. [141]. The authors focus on renormalizable H∗ H H models with combinations of and (or ) in the Yukawa sector. They work out, neglecting the first generation masses, some interesting analytic correlations 126 10 120 between the neutrino and the charged fermion sectors. In a recent paper [221] the full three generation study of such settings has been H H∗ numerically addressed. The authors claim that the model with ⊕ cannot fit H H∗ the fermions, while the setting with and yields an excellent fit in the case of 120 126 type-I seesaw dominance. 10 126 H A subtle feature, as pointed out in [141],m ist thatm theb must be complex. The reason being that in the real case one predicts ∼ (at least when working in the two 10 H heaviest generations limit and with real parameters). A complex implies then the presence of one additional Yukawa coupling. In turn this entails a loss of predictivity 10 in the Yukawa sector when compared to the supersymmetric case. The proposed way out advocated by the authors of Ref. [141] was to consider a PQ symmmetry, relevant for dark matter and theSO strong CP problem, which forbids that extra Yukawa. Sticking to a pure approach, some predictivity could be also recovered working with three Yukawas but requiring only one Higgs doublet in the effective (10) theory, as a preliminary numerical study with three generations shows [151]. H H H The comparison between the ⊕ ⊕ scenario and the next-to-minimal H H H one with a in place of a is also worth a comment. At first sight the seems 10 45 126 a good option as well in view of the two-loop values emerging from the unification 54 MB L 45 : MU : 54 analysis of Chapter 2: − × GeV and × GeV. However the H H 13 15 choice between the and the leads to crucially distinctive features. =4 7 10 =1 2 10 H∗ The first issue has to do with the nature of the light Higgs. In this respect the 54 45 plays a fundamental role in the Yukawa sector where it provides the necessary breaking 126 of the down-quark/charged-lepton mass degeneracy (cf. Eqs. (1.181)–(1.182)).; For; this toH work one needs a reasonably large admixture between the bi-doublets ⊂ ; ; H∗ H and ⊂ . In the model with the this mixing is guaranteed by the H H∗ H H (1 2 2) 10 interaction , but there is not such a similar invariant in the case of the H (15 2 2) 126 45 H H∗ H H . Though there always exists a mixing term of the type , this 10 126 45 45 H H yields a suppressed mixing due to the unification constraint h i≫h i. 54 H 10 126H 126 126 The other peculiar difference between the models with and has to do with 45 126 the interplay between type-I and type-II seesaw. As already observed in Sect. 1.5.3 one 45 54 SUexpectsR that in theories in which the breaking of the D-parity isM decoupledB L/MU from that of the type-II seesaw is naturally suppressed by a factor − with respect H H 2 to the type-I. Whilst the leads to this last class of models, the preserves the D- (2) H ( SU R ) parity which is subsequently broken by the together with . The dominance 45 H 54 of type-I seesaw in the case of the has a double role: it makes the Yukawa sector b τ126 (2) more predictive and it does not lead to - unification, which is badly violated without 45 supersymmetry. Outlook: the quest for the minimal nonsupersymmetric SO theory 151 (10)

So where do we stand at the moment?H H In orderH to say something sensible one has to test the consistency of the ⊕ ⊕ vacuum against gauge unification and the SM fermion spectrum. If the vacuum turned out to be compatible with the 10 45 126 phenomenological requirements it would be then important to perform an accurate estimate of the proton decay braching ratios. As a matter of fact nonsupersymmetric GUTs offer the possibility of making definite predictions for proton decay, especially in H H H the presence of symmetric Yukawa matrices, as in the ⊕ ⊕ case, where the main theoretical uncertainty lies in the mass of the leptoquark vector bosons, subject 10 45 126 to gauge unification constraints. Though the path is stillSO long we hope to have contributed to a little step towards the quest for the minimal theory. (10) Outlook: the quest for the minimal nonsupersymmetric SO theory 152 (10) Appendix A

One- and Two-loop beta coeﬃcients

β In this appendix we report the one- and two-loopU -coeﬃcients used in the numerical analysis of Chapter 2. The calculation of the mixing coeﬃcients and of the Yukawa contributions to the gauge coupling renormalization is detailed in Apps. A.1 and A.2 (1) respectively.

MZ M Ï ai bij SM ( 1) Chain − ; ; − −  9 11  26 2 10 19 41  35 9  All ( 7 6 10 ) 12 6 10  44 27 199  5 10 50

ai bij C L Y Table A.1: The and coeﬃcients are given for the (SM) gauge running. The scalar sector includes one Higgs doublet. 3 2 1 Appendix A. One- and Two-loop beta coefficients 154

M M Ï ai bij ai bij G1 ( 1 2)

Chain − Chain −  9 9 1   9 9 1  ; ; ; 2 2 2 ; ; ; 2 2 2 − − − 26 − − − 26  3   3  7 11  12 8 3 2  17 17  12 8 3 2  Ia ( 7 3 3 2 )  80 27  Ib ( 7 3 6 4 )  61 9   12 3 3 2   12 3 6 4   9 81 61   9 27 37  2 2 2 2 4 8 − 4 −4  9 9 1   9 9 1  ; ; ; 2 2 2 ; ; ; 2 2 2 − − − 26 − − − 26  80 27   61 9  7 7  12 3 3 2  17 17 9  12 6 3 4  IIa ( 7 3 3 7)  80 27  IIb ( 7 6 6 2 )  61 9   12 3 3 2   12 3 6 4   81 81 115   27 27 23  2 2 2 4 4 4 −4 −4  9 9 1   9 9 1  ; ; ; 2 2 2 ; ; ; 2 2 2 − − − 26 − − − 26  3   3  7 11  12 8 3 2  17 17  12 8 3 2  IIIa ( 7 3 3 2 )  80 27  IIIb ( 7 3 6 4 )  61 9   12 3 3 2   12 3 6 4   9 81 61   9 27 37  2 2 2 2 4 8 −4 −4  9 9 1   9 9 1  ; ; ; 2 2 2 ; ; ; 2 2 2 − − − 26 − − − 26  3   3  7 11  12 8 3 2  17 17  12 8 3 2  IVa ( 7 3 3 2 )  80 27  IVb ( 7 3 6 4 )  61 9   12 3 3 2   12 3 6 4   9 81 61   9 27 37  4 2 2 2 4 2 4 8 − − ; ; ; ; − −  101 9 27  − −  295 9  6 2 2 4 2 2 29 19 15  45 35 1  21 19 9  45 35 1  Va ( 3 6 2 )  2 6 2  Vb ( 2 6 2 )  2 6 2  405 3 87 3 9 2 2 2 30 2 2 − − ; ; ; ; − −  101 9 27  − −  295 9  6 2 2 4 2 2 29 19 15  45 35 1  21 19 9  45 35 1  VIa ( 3 6 2 )  2 6 2  VIb ( 2 6 2 )  2 6 2  405 3 87 3 9 2 2 2 30 2 2 − ; ;  643 9 153  ; ;  206 9 15  − − 6 2 2 − − − 3 2 2 23 11  45  31 7  45  VIIa ( 3 3 3 )  2 8 3  VIIb ( 3 3 3 )  2 8 3  765 584 75 50 2 3 3 2 3 3 ai bij The and coeﬃcients due to gauge interactions are reported for the G1 chains I to VII Table A.2:H H with (left) and (right) respectively. The two-loop contributions induced by Yukawa couplings are given126 in Appendix16 A.2 155

M M Ï ai bij ai bij G1 ( 1 2) Chain Chain −26 −26  9 3 1  VIIIb  9 3 1  ; ; ; 12 2 2 2 ; ; ; 12 2 2 2 − − − − VIIIa ( 7 )  35 1 3  ( 7 )  35 1 3  . 19 9 9  12 6 2 2  . 19 17 33  12 6 2 2  . 6 2 2  3 15 15  . 6 4 8  3 15 15  XIIa.  4 2 2 2  XIIb.  4 2 4 8   9 15 25   9 15 65  2 2 2 2 8 16

ai bij Table A.3: The and coefficients due to purely gaugeβ interactions for the G1 chains VIII to XII are reported. For comparisonU with previous studies the -coefficientsU are given neglecting systematically one- and two-loops mixing effects (while all diagonal contributions to abelian andU non-abelian gauge coupling renormalization(1) are included). The complete(1) (and correct) treatment of mixing is detailed in Appendix A.1. (1) Appendix A. One- and Two-loop beta coefficients 156

M MU Ï aj bij aj bij G2 ( 2 ) Chain Chain − ; ;  289 9 153  ; ;  94 9 15  (−7 −3 ) 2 82 32 (− −3 − ) 3 82 32 11  45  29 7  45  Ia 3 2 3 Ib 3 3 2 3  765 584   75 50  2 3 2 3 − ; ;  661 153 153  ; ;  127 15 15  (−4 ) 2 2 32 (− − − ) 6 2 32 11 11  765 584  28 7 7  75 50  IIa 3 3 2 3 IIb 3 3 3 2 3  765 584   75 50  2 3 2 3 − ; ;  661 153 153  ; ;  127 15 15  (−4 ) 2 2 32 (− − − ) 6 2 32 11 11  765 584  28 7 7  75 50  IIIa 3 3 2 3 IIIb 3 3 3 2 3  765 584   75 50  2 3 2 3 −26 −26  9 9 1   9 9 1  ; ; ; 12 2 32 2 ; ; ; 12 2 32 2 − − − − − − ( 7 7)  80 27  ( 7 )  61 9  7 7  12 3 2  17 17 9  12 36 4  IVa 3 3  80 27  IVb 6 6 2  61 9   4 3 2   4 6 4   81 81 115   27 27 23  2 2 2 4 4 4 − ; ;  643 9 153  ; ;  206 9 15  (− −3 4) 6 82 32 (− −3 −2) 3 82 32 23  45  31  45  Va 3 2 3 204 Vb 3 2 3 26  765   75  2 2 − ; ;  1759 153 153  ; ;  117 15 15  (− 4 4) 6 2042 32 (−10 −2 −2) 2 262 32 14  765   75  VIa 3 2 3 204 VIb 2 3 26  765   75  2 2 − ; ;  1759 153 153  ; ;  117 15 15  (− ) 6 2 32 (−10 − − ) 2 2 32 14 11 11  765 584  7 7  75 50  VIIa 3 3 3 2 3 VIIb 3 3 2 3  765 584   75 50  2 3 2 3 −26 −26  9 9 1   9 9 1  ; ; ; 12 82 32 2 ; ; ; 12 82 32 2 − − − − − − ( 7 3 2 )  3  ( 7 3 )  3  11  12 3 36 2  5 17  12 3 2  VIIIa 2  27  VIIIb 2 4  39 9   4 2   4 2 4   9 81 61   9 27 37  2 2 2 2 4 8 −26 −26  9 9 1   9 9 1  ; ; ; 12 362 32 2 ; ; ; 12 2 32 2 − − − − − − IXa ( 7 2 2 7)  27  IXb ( 7 )  39 9   12 3 36 2  5 5 9  12 32 4   27  2 2 2  39 9   4 2   4 2 4   81 81 115   27 27 23  2 2 2 4 4 4

; ;  1315 9 249  ; ;  130 9 111  Xa (− −3 ) 6 82 32 Xb (− −3 ) 3 82 32 17 26 45 25 8 45 3 3  2 3  3 3  2 3   1245 1004   555 470  2 3 2 3

; ;  3103 249 249  ; ;  331 111 111  XIa (− ) 6 2 32 XIb (−6 ) 2 2 32 2 26 26  1245 1004  8 8  555 470  3 3 3  2 3  3 3  2 3  1245 1004 555 470 2 3 2 3 − 2 ; ;  41 9 27  ; ;  437 9  XIIa (−9 − ) 2 2 2 XIIb (− − ) 12 2 19 15 45 35 1 59 19 9 45 35 1 6 2  2 6 2  6 6 2  302 6 2   405 3 87   3 9  2 2 2 2 2

ai bij Table A.4: The and coeﬃcients due to pure gauge interactions are reported for the G2 chains H H with (left) and (right) respectively. The two-loop contributions induced by Yukawa couplings are given126 in Appendix16 A.2 157

bij Chain ˜ Eq. in Ref. [100]

− −  199 81 44  All/SM 205 − 95 − 35 A7  9 35 12   41 − 19 7  11 27 26 41 19 7 − −  25 5 27 4  9 3 − 19 − 7 VIIIa/G1  5 5 9 12  A10  3 3 − 19 − 7   1 1 35 12   3 9 − 19 7   1 1 27 26  9 3 19 7 − − −  61 3 81 4  11 − 2 − 4 − 7 VIIIa/G2  3 8 3 12  A13  11 −13 −182 − 7   27 12   11 − − 7   1 3 9 26  11 2 4 7 − −  8 9 45  Ia/G2 −13 11 − 14 A14 584 765  − 11 − 14   3 459 289  2 22 14 − −  35 1 135  Va/G1 − 19 15 − 58 A15 9 29 1215  − 19 5 58   27 9 101  19 5 58 − −  35 1 5  XIIa/G2 − 19 15 − 2 A18 9 29 45  − 19 5 − 2   27 9 41  19 5 18

β bij Table A.5: The rescaled two-loop -coeﬃcients computed in this work are shown together with the correspondingU equations in Ref. [100]. For the˜ purpose of comparison Yukawa contributions are neglected and no mixing is included in chain VIIIa/G1. Care must be taken of the diﬀerent ordering between abelian(1) and non-abelian gauge group factors in Ref. [100]. We report those cases where disagreement is found in some of the entries, while we fully agree with the Eqs. A9, A11 and A16. Appendix A. One- and Two-loop beta coefficients 158

φ ai bij

126

48 48 ; ; ; ;  896  (15 2 2) ( 5 5) 2403 65 45 16   3  240 45 65 

24 8 ; ; ; ;  448  (15 2 + )( ) 1203 1 8 5 5 65 15 2 3 2 2  120 2 2   45 15  2 2 β One- and two-loop additional contributions to the -coeﬃcients related to the presence of Tableφ A.6: C L R C L R the scalar multiplets in the (top) and (bottom) stages. 126 4 2 2 4 2 1 A.1. Beta-functions with U mixing 159 A.1 Beta-functions(1) with U mixing

β The basicU building blocks of the one- and two-loop(1) -functions for the abelian couplings with mixing, cf. Eqs. (2.13)–(2.14), can be conveniently written as (1) gkagkb gsa sr(1)grb (A.1) = Γ and

gkagkbgkc2 gsa sr(2)grb ; (A.2)

= Γ a (1) (2) Qk where and are functions of the abelian charges and, at two loops, also of U A the gauge couplings. In the case of interest, i.e. for two abelian charges and U B Γ Γ , one obtains (1)

(1) A AA(1) Qk 2 ;

A B AB(1) BA(1) Qk Qk ; Γ = ( ) (A.3) QB ; ΓBB(1) =Γ k 2=

Γ = ( ) and

A A B A B AA(2) Qk 4 gAA2 gAB2 Qk 3Qk gAAgBA gABgBB Qk 2 Qk 2 gBA2 gBB2 ;

A B A B A B ΓAB(2) =( BA(2) ) ( Qk+3Qk g)+2(AA2 gAB2) ( Qk 2 Q+k 2 gAAgBA)+( gAB) g( BB ) ( Qk +Qk 3 )gBA2 gBB2 ;

A B A B B ΓBB(2) =ΓQk 2=(Qk 2)gAA2 ( gAB2+ Q)+2(k Qk 3)g(AAg)BA( gABg+BB Qk)+4 gBA2 ( gBB)2 ( : + )

Γ =( ) ( ) ( + ) + 2 ( ) ( + )+( ) ( + ) (A.4) All other contributions in Eq. (2.13) and Eq. (2.14) can be easily obtained from Eqs. (A.3)– (A.4) by including the appropriateA n B m group factors. It is worth mentioning that for com- SO Qk Qk n m pleten m multiplets, for n and m odd (with at one-loop and at two-loop level). (10) ( ) ( ) =0 + =2 C L R B L By+ evaluating=4 Eqs. (A.3)–(A.4) for the particle content relevant to the − stages in chains VIII-XII, and by substituting into Eqs. (2.13)–(2.14), one ﬁnally obtains 3 2 1 1 Appendix A. One- and Two-loop beta coefficients 160

H • Chains VIII-XII with in the Higgs sector: 126

γC gR;R2 gR;B2 L gB2 L;R gB2 L;B L gL2 gC2 ; − π − − − − − 2 1 3 1 9 = 7+ ( + )+ ( + )+ 26 (A.5) γL (4 ) 2 gR;R2 gR;B2 L 2 gB2 L;R gB2 L;B L 2 gL2 gC2 ; − π − − − − 2 19 1 1 3 35 = + ( + )+ √( + )+ + 12 γR;R 6 (4 ) 2 gR;R2 gR;B2 L 2 gR;RgB L;R gR;B L6gB L;B L π − − − − − − 2 9 1 15 = + ( + ) 4 6( + ) 2 (4 )gB2 L;R2 gB2 L;B L gL2 gC2 ; − − − 15 3 + ( + )+ √ + 12 γR;B L γB 2L;R 2 gR;R2 gR;B2 L − − −√ π 2 − − 1 1 h = = + 2 6( + √ ) gR;RgB L;R gR;B LgB L;B L gB2 L;R gB2 L;B L ; −6 (4 ) − − − − − − − i + 15( + ) √ 3 6( + ) γB L;B L gR;R2 gR;B2 L gR;RgB L;R gR;B LgB L;B L − − π − − − − − 2 − 9 1 15 = + ( + ) 6 6( + ) 2 g(4B2 L;R) 2gB2 L;B L gL2 gC2 − − − 25 9 + ( + )+ + 4 ; 2 2 A.2. Yukawa contributions 161

H • Chains VIII-XII with in the Higgs sector:

γC 16gR;R2 gR;B2 L gB2 L;R gB2 L;B L gL2 gC2 − π − − − − − 2 1 3 1 9 = 7+ ( + )+ ( + )+ 26 (A.6) γL (4 ) 2 gR;R2 gR;B2 L 2 gB2 L;R gB2 L;B L 2 gL2 gC2 ; − π − − − − 2 19 1 1 3 35 = + ( + )+ ( + )+ + 12 γR;R 6 (4 ) 2 gR;R2 gR;B2 L 2 gR;RgB L;R gR;B6 LgB L;B L π − − − − 2 " − − r 17 1 15 1 3 = + ( + ) ( + ) 4 (4 g)B2 L;R4 gB2 L;B L gL22 2 gC2 ; − − − 15 3 + ( + )+ + 12 γR;B L γB 8L;R 2 gR;R2 gR;B2 L − − √ π − 2 "− r − 1 1 1 3 = = + ( + ) 4 6 (4 ) 4 2 gRRgB L;R gR;B LgB L;B L gB2 L;R gB2 L;B L ; − − − − − r − − − # 15 3 3 + ( + ) ( + ) 4 8 2 γB L;B L gR;R2 gR;B2 L gRRgB L;R gR;B LgB L;B L − − π − − − − 2 " − − r 33 1 15 3 3 = + ( + ) ( + ) 8 gB2(4L;R) g8B2 L;B L gL2 g4C2 2: − − − 65 9 + ( + )+ + 4 16 2 γB L;R γR;B L gB L;R gR;B L By setting − − β and − − in Eqs. (A.5)–(A.6) one obtains the one- and two-loop -coeﬃcients in the diagonal approximation, as reported = = 0 = = 0 in Table A.3. The latter are used in Figs. 2.1–2.2 for the only purpose of exhibiting the eﬀect of the abelian mixing in the gauge coupling renormalization.

A.2 Yukawa contributions

β The Yukawa couplings enter the gauge -functions first at the two-loop level, cf. Eq. (2.2) and Eq. (2.13). Since the notation adopted in Eqs. (2.5)–(2.6) is rather concise we shallypk detail the structure of Eq. (2.5), paying particular attention to the calculation of the coefficients in Eq. (2.19). The trace on the RHS of Eq. (2.5) is taken over all indices of the fields entering the

Yukawa interaction inQL Eq.YU U (2.6).Rh Consideringh:c: h foriσ instanceh∗ the up-quark Yukawa sector of the SM the term (with 2 ) can be explicitly written as ab kl i a bj ˜ Y+U ε δ j QL ikU˜ R=hl∗ h:c:; 3 (A.7) + Appendix A. One- and Two-loop beta coefficients 162

a;b i;j k; l SU C SU L { } { } { } whereδn , andn label flavour, δ and indices respectively, while denotes the -dimensional Kronecker symbol. Thus, the Yukawaab kl couplingi (3) (2) YU ε δ j entering Eq. (2.5) is a 6-dimensional object with the index structure 3 . The ypU contribution of Eq. (A.7) to the three coefficients (convenientlyQL UR separated into two terms corresponding to the fermionic representations and ) can then be written as p p ab kl i ab j ypU C( ) QL C( ) UR YU ε δ j YU ∗εklδ i d Gp 3 3 2 2 ab;ij;kl (A.8) 1 h i X = ( ) + ( ) ab ab ( ) ab YU ∗YU YU YU† The sum can be factorized into the flavour space part kl i times † ε δ j the trace over the gauge contractions where P≡ 3 . For the SM gauge =y Tr[U 17 y] U 3 group (with the properly normalized hypercharge) one then obtains 1 10 , 2 2 y U Tr[∆∆ ] ∆ and 3 , that coincide with the values given in the first column of the matrix (B.5) = = in Refs. [182, 183, 184]. =2 ypk All of the coefficients as well as the structures of the relevant -tensors are reported in Table A.7. ∆ A.2. Yukawa contributions 163

Gp ypk k † Gauge structure Higgs rep. Tensor ∆ Tr[∆∆ ]

Q Ui h klδ j C Lkj R ˜l i 3 2 2 0 U i l hl ; ; k j 6 L   QLkj DRh δ l δ i 2 D i l : (+ 2 1) k3 6 Y 3 3 1 LLkERh 1 δ l 1  2 2 2  E 2 2 3 2  17 1 3  10 2 2 2 C 3 2 2 0 0 i kl j L   QL U hl δ 2 kj R ˜ i 3 3 1 1 U i l k j 6 R;R QLkj DRh l δ l δ i  2 2 2 2  h ; ; ; 3 1  3 3 1 1  D kl 6 R;B L   LLkNRhl : (2 + 0 1) −  2 − 2 − 2 2  N ˜ 1 2 3 2 1  q q q q  L E hl δ k B L;R  1 3 1 3 1 3 1 3  Lk R 2 l 1 −  2 2 − 2 2 − 2 2 2 2  E 2 B L;B L  1 q 3 1 q 3 1 q 3 1 q 3  − −   2 1  2 2 2 2 2 2 2 2  1 1 3 3 2 2 2 2 C 3 4 0 j QikQc mφln δ L   L Lj ln kl mn i 2 3 1 Q k c m ln φ ; ; ; 12 R LLLL φ : (2 2 0 1) klmn   3 2  3 1  L 4 B L   1 −  1 3 

C F F Uih klδ j 4 2 2 Lkj R l hl ; ; i L   F Di ˜l k j 8 2 2 2 U FLkj FR h : (2 + 1) δ l δ i R F 1 4 8 1  2 2  D 2   2 4 C 4 4 F ikF c mφln φln ; ; j L   L Lj klmnδ i 2 4 F : (2 2 1) 16 R   2  4  4

C F F UiHa kl T j 4 Lkj R l Hla ; ; a i L  15 15  F Di ˜la k ( )j 15 2 4 4 U FLkj FR H : (2 + 15) δl Ta i R 15 15 F 1 ( ) 15 1  4 4  D 2  15 15  4 4 C 4 ik c m lna lna j L  15  FL FLj ; ; klmn Ta i 2 2 F Φ Φ : (2 2 15) ( ) 30 R 15 2  2   15  2 β The two-loop Yukawa contributions to the gauge sector -functions in Eq. (2.19) are detailed. Table A.7:p ypk k The index in labels the gauge groups while refers to flavour. In addition to the Higgs bi-doublet from the 10-dimensional representation (whose components are denoted according to the relevant gauge h φ H H symmetry by and ) extra bi-doublet components in (denoted by and ) survives from unification down to the Pati-Salam breaking scale as required by a realistic SM fermionic spectrum. The Ta SU C 126 Φ factors are the generators of in the standard normalization. As a consequence of minimal H H SU C fine tuning, only one linear combination of and doublets survives below the scale. The U R;B L C L R (4)B L − − mixing in the case is10 explicitly126 displayed. (4) (1) 3 2 1 1 Appendix A. One- and Two-loop beta coefficients 164 Appendix B

SO algebra representations

SO We brieﬂy(10) collect here the conventions for the algebra representations adopted in Chapter 3. (10) B.1 Tensorial representations

SOThe hermitian and antisymmetric generators of the fundamental representation of are given by ij ab i δa iδbj ; − [ ] (B.1) (10)a;b;i;j ;::; where SO and the square( ) = bracket( stands) for anti-symmetrization. They satisfy the commutation relations = 1 10 ij ;kl i δjkil δikjl δjlik δiljk ; (10) − − − (B.2) with normalization = ( + ) ij kl δi kδjl : Tr [ ] (B.3) φa a ; :::; The fundamental (vector) representation =2 transforms as i φa φa (λij =1ij φ a ; 10) Ï − (B.4)

λij ( ) where are the inﬁnitesimal parameters of2 the transformation. The adjoint representation is then obtained as the antisymmetric part of the 2-index a b φab a;b ;::; ⊗ tensor and transforms as i 10 10 ( =1φab 10) φab λij ij ; φ : Ï − ab (B.5) T ij ; φ ij ; φ ij ; φ † ij ; φ Notice that − and 2 . = = Appendix B. SO algebra representations 166 B.2 Spinorial representations (10)

SO Sij i;j ;::; Following the notation of Ref. [59], the generators ( ) acting on the 32-dimensional spinor are deﬁned as (10) = 0 9 Ξ Sij i; j ; i (B.6) 1 i = Γ Γ where the ’s satisfy the Cliﬀord algebra4 i; j δij : Γ { } (B.7)

An explicit representation given by [222]Γ Γ =2

I isp 16 ; p ; p ; :::; ; 0 I ! isp ! (B.8) 16 − 0 0 spΓ = Γ k= ;::; =1 9 where the matrices are0 deﬁned as ( ) 0 sk ηkρ ; sk σkρ ; sk τkρ : 3 +3 =1 1 3 +6 2 (B.9) σk τk ηk ρk The matrices , , =and , are given= by the following= tensor products of × matrices 2 2 σk I I I k ; 2 ⊗ 2 ⊗ 2 ⊗ τk I I k I ; 2 ⊗ 2 ⊗ ⊗ 2 (B.10) ηk = I k I ΣI ; 2 ⊗ ⊗ 2 ⊗ 2 ρk = k I ΣI I ; ⊗ 2 ⊗ 2 ⊗ 2 = Σ k where stand for the ordinary Pauli=Σ matrices. Deﬁning

Σ spq sp;sq i (B.11) 1 p; q ;::; = for , the algebra (B.6) is represented2 by sp spq =1 9 Sp ; Spq : 0 sp spq (B.12) − ! ! 1 0 1 0 = S S S= S S The Cartan subalgebra is2 spanned0 over 03, 12, 452, 78 and0 69. One can construct a χ chiral projector , that splits the 32-dimensional spinor into a pair of irreducible 16-dimensional components: Γ Ξ I χ −5S S S S S − 16 : 03 12 45 78 69 I ! (B.13) 016 Γ =2 = 0 B.3. The charge conjugation C 167

χ χ2 I χ; i It is readily verified that has the following properties: 32, { } and χ; Sij P 1 I χ ± 32 ∓ hence . IntroducingΓ the chiral projectors Γ2 = ,Γ theΓ irreducible= 0 chiral spinors are defined as Γ =0 = ( Γ ) χ χ P ; χ P ; c + + ≡ ! − − ≡ χ ! (B.14) 0 c = Ξ = Ξ χ Cχ∗ C SO ≡ 0 where and is the charge conjugationSij matrix (see next subsection). Analogously, we can use the chiral projectors to write as (10) σij Sij P Sij P P Sij P ; + + − − ≡ σij ! (B.15) 1 0 = + P ; Sij P2 P P2 0P ˜ I where the properties ± , ± ± and + − 32 were used. Finally, matching Eq. (B.15) withc Eq. (B.12), one identifies the hermitian generators σij / σij / χ =0 χ = + = and acting on the and spinors, respectively, as

σp sp ; σpq spq ; σp sp ; σpq spq ; 2 ˜ 2 0 0 − (B.16) with normalization = = ˜ = ˜ = 1 σij σkl 1 σij σkl δi kδjl : 4 Tr 4 Tr [ ] (B.17) σ It is convenient to trace out the -matrices in the invariants built oﬀ the adjoint = σ˜ij φ˜ij / =4 σrepresentation in the natural basis ≡ . From the traces of two and four -matrices one obtains Φ 4 2 φ2 ; Tr − Tr (B.18) φ 2 φ : Φ4 = 3 2 2 4 Tr 4 Tr − Tr (B.19) In order to maintain a consistentΦ = notation, from now on we shall label the indices of the spinorial generators from 1 to 10, and use the following mapping from the basis of Ref. [59] into the basis of Ref. [51] for both vectors and tensors: { }Ï { }. 0312457869 B.312345678910 The charge conjugation C

χ Accordingχ to∗ the notation of the previous subsection, the spinor and its complex conjugate transform as i i T χ χ λij σij χ; χ∗ χ∗ λij σij χ∗ : Ï − Ï (B.20) + 4 4 Appendix B. SO algebra representations 168

c χ Cχ∗ (10) The charge conjugated spinor ≡ obeys c c i c χ χ λij σij χ ; Ï − (B.21) C ˜ and thus satisﬁes T C−1σij C 4 σij : − (B.22) Taking into account Eq. (B.10), a formal˜ solution= reads C σ τ η ρ ; 2 2 2 2 (B.23) which in our basis yields = C ; ; ; ; ; ; ; ; antidiag − − − − ; ; ; ; ; ; ; ; − − − − = (+1 1 1 +1 1 +1 +1 1 (B.24) T C C∗ C−1 C C† and hence . 1 +1 +1 1 +1 1 1 +1) B.4 The= Cartan= = generators=

SO C L R B L − It is convenient to writeTB L the ﬁveB L / Cartan generators in the basis, where the generator − is − . For the spinorial representation we have (10) 3 2 2 1 TR3 1 σ σ ; TR3 1 σ σ ; (4 12 ) 234 4 − 12 34 TL3 1 σ σ ; TL3 1 σ σ ; = 4( 34+− 12) e = 4( 34 + 12 ) Tc3 Tc3 1 σ σ ; = ( ) 4 e56 =− 78( + ) Tc8 Tc8 √1 σ σ σ ; = e4 =3 56( 78 − ) 910 TB L TB L 2 σ σ σ : − − = e = −( 3 +56 78 2 910) (B.25) c T χ T= e = ( + + ) σ i χ While the ’s act on , the ’s (characterized by a sign ﬂip in 1 ) act on . The normalization of the Cartan generatorse is chosen according to the usual SM convention.TB L A GUT-consistent normalization across all generators is obtained by rescaling − (and TB L √ / − ) by . In order to obtain the physical generators acting on the fundamental representation e 3 2 σij / ij it is enough to replace in Eq. (B.25) by . χ χc With this information at hand, one can identify the spinor components of and 2 χ ν;u ;u ;u ;l;d ;d ;d ; dc;dc;dc; lc;uc; uc; uc;νc ; 1 2 3 1 2 3 − 3 2 1 − 3 − 2 − 1 (B.26)

c = ( c c c c c c c c ) and χ ν ;u ;u ;u ; l ;d ;d ;d ; d ;d ;d ; l;u ; u ; u ;ν ∗ ; 1 2 3 1 2 3 − 3 2 1 − 3 − 2 − 1 (B.27) = ( ) B.4. The Cartan generators 169 where a self-explanatory SM notation has been naturally extended into the scalar sector. c In particular,SO theSU relativeC signsSO in Eqs.SU (B.26)–(B.27)L SU R arise from the chargeχ conjugationχ of the ∼ and ∼ ⊗SU components of and . The standard and ﬂipped embeddings of commute with two diﬀerent Cartan (6)Z (4)Z′ (4) (2) (2) generators, and respectively: (5) Z TR(3) TX ; Z′ TR(3) TX : − (B.28) T T R3 =2 43 +6B2 L =4 +6 Given the relation Tr 2 Tr − one obtains YZ ; YZ′ ; ( ) =Tr Tr 6 (B.29)

Y TR3 TB L − ( )=0 ( ) =0 where is the weak hyperchargeSU generator. SU ′ As a consequence,SU SU the standardU contains the SM group, while has a = +C L Y ′ subgroup ⊗ ⊗ , with (5) (5) Y ′ TR3 TX : (3) (2) (1) − (B.30) Z′ Y ′ In terms of and of the weak hypercharge= + reads

Y 1 Z′ Y ′ : 5 − (B.31)

Using the explicit form of the Cartan generators= ( in) the vector representation one ﬁnds Z′ ; ; ; ; ; ∝ diag − − ⊗ 2 (B.32) Z ; ; ; ; : ∝ diag( 1 1 +1 +1 +1) ⊗ Σ2 (B.33)

ωR ωB L ωR ωB L −(+1− +1 +1 +1 +1)− Σ ZThe′ vacuumZ conﬁgurations and SUin Eq.′ U (3.7)Z are alignedSU withU theZ and the generator respectively, thus preserving ⊗ ′ and ⊗ , = = respectively. (5) (1) (5) (1) Appendix B. SO algebra representations 170 (10) Appendix C

Vacuum stability

The boundedness of the scalar potential is needed in order to ensure the global stability of the vacuum. The requirement that the potential is bounded from below sets non trivial constraints on the quartic interactions. We do not provide a fully general analysis for the whole ﬁeld space, but limit ourselves to the constraints obtained for the given vacuum directions. ωR ωB L χR • , − , 6 V (4) From( the quartic) =0 part of the scalar potential 0 one obtains a λ a ωR2 ωB2 L 2 2 ωR4 ωB4 L ωR2 ωB2 L 1 χR4 αχR2 ωR2 ωB2 L 1 − − − − β τ 4 (2 +3 ) + (8 +21χR2 ωR+36ωB L 2 ) +χR2 ωR+4 ωB(2L >+3 ) 4 − − 4 − (C.1) λ + (2 +3 H ) (2 +3 ) 0 Notice that the 2 term vanishes4 along the vacuum2 direction. ωR ωB L χR • − , 6 16 V (4) Along= this=0 direction=0 the quartic potential 0 reads

V (4) 1 λ χR4 ; 0 4 1 (C.2)

which implies λ=> : 1 (C.3) χR From now on, we focus on the case, cf.0 Sect. 3.3.6. ω ωR ωB L χR • − − , =0 On= this= orbit the quartic=0 part of the scalar potential reads V (4) 5 ω4 a a : 0 4 1 2 (C.4) = (80 + 13 ) Appendix C. Vacuum stability 172

a < Taking into account that the scalar mass spectrum implies 2 , we obtain a > a : 13 0 1 − 80 2 (C.5)

ωR ωB L χR • , − 6 , At the=0 tree level=0 this=0 VEV conﬁguration does not correspond to a minimum of the potential. It is nevertheless useful to inspect the stability conditions along this direction. Since V (4) 3 a a ωB4 L ; 0 4 1 2 − (C.6) a boundedness is obtained, independently= (48 on+7 the) sign of 2, when

a > 7 a : 1 − 48 2 (C.7)

ωR ωB L χR • 6 , − , In analogy=0 with=0 the previous=0 case we have

V (4) a a ωR4 ; 0 1 2 (C.8)

which implies the constraint = 2(8 + ) a > 1 a : 1 − 8 2 (C.9) a < Ina the case 2 the constraint in Eq. (C.5) provides the global lower bound on 1. 0 Appendix D

Tree level mass spectra

D.1 Gauge bosons

Given the covariant derivatives of the scalar ﬁelds Dµφ ab ∂µφab i g Aµ ij ij ; φ ; − ab (D.1) 1 ( ) = ( ) 2 Dµχ α ∂µχα i g Aµ ij σij αβ χβ ; − (D.2) 1 ( ) = ( ) ( ) c c 4 c Dµχ α ∂µχα i g Aµ ij σij αβ χβ ; − (D.3) 1 ( ) = ( ) (˜ ) and the canonically normalizaed kinetic terms4 µ Dµφ † D φ ; Tr (D.4) 1 and µ ( ) ( c) µ c Dµχ † 4D χ Dµχ † D χ ; (D.5) 1 1 one may write the ﬁeld dependent( ) ( mass) +matrices( for) ( the) gauge bosons as 2 2 g 2 A2 φ ij kl ij ; φ kl ; φ ; M ( )( ) Tr ( ) ( ) (D.6) g 2 A2 (χ) ij kl = χ† [σ ij ;σ ][kl χ: ] M ( )( ) 2 { ( ) ( )} (D.7) ij ; kl σ / i;j ;::; ( ) = ij ij where stand for ordered pairs of4 indices, and ( ) with are the generators of the fundamental (spinor) representation (see Appendix B). ( ) ( ) ωR;B L ; χR 2 = 1 10 Eqs. (D.6)–(D.7), evaluated on the generic ( − 6 6 ) vacuum, yield the following contributions to the tree level gauge boson masses: = 0 = 0 Appendix D. Tree level mass spectra 174 D.1.1 Gauge bosons masses from 45

Focusing on Eq. (D.4) one obtains ; ; g ω ; MA2 2 R2 A2 ; ; 2 g 2ωB2 L ; M (1 1 −+1)3 = 4 − A2 ; ; ; M (3 1 )=4 A2 ; ; ; M (1 3 0)=0 (D.8) A2 ; ; 5 g 2 ωR ωB L 2 ; − M (8 1 −0)=06 − A2 ; ; 1 g 2 ωR ωB L 2 ; M (3 2 6 ) = ( − ) A2 (3; 2; + ) = ( + ; ) M ! 0 0 (1 1 0) = 0 0 ψ45 ψ45 15 1 where the SM singlet matrixSO is deﬁned on the basis ( , ), with theSU superscriptC referring to the original representation and the subscript to the origin (see Table 4.5). ω ω ω ω (10) Z R B L ′ Z′ R (4) B L − − − C NoteL R B that,L ω inR the limitsC ofL standardR ωB L , ﬂipped , − ( ) and ( − ) vacua, we have respectively 25, 25, 15 and 51 ( = ) 5 1 ( = ) 19 massless gauge bosons, as expected. 3 2 2 1 =0 4 2 1 =0 D.1.2 Gauge bosons masses from 16

The contributions from Eq. (D.5) read ; ; g χ ; MA2 2 R2 A2 ; ; 2 g 2χR2 ; M (1 1 +1)− 3 = A2 ; ; ; M (3 1 ) = A2 ; ; ; M (1 3 0)=0 (D.9) A2 ; ; 5 ; M (8 1 −0)=06 A2 ; ; 1 g 2χR2 ; M (3 2 6)=0 3 3 (3; 2; + ) = 2 2 g χ ; MA2  q  2 R2 3 2 (1 1 0) =  q  ψ45 1ψ45 where the last matrix is again spanned over ( 15, 1 ), yielding A2 ; ; ; Det M (D.10) A2 ; ; 5 g 2χR2 : Tr M (1 1 0)=02 (D.11) (1 1 0) = D.2. Anatomy of the scalar spectrum 175

SU The number of vanishingH entriesχR corresponds to the dimension of the algebra preserved by the VEV . H H (5) Summing together the and contributions, we recognize 12 massless states, 16 that correspond to the SM gauge bosons. 45 16 D.2 Anatomy of the scalar spectrum

In order to understand the dependence of the scalar masses on the various parameters in the Higgs potential we detail the scalar mass spectrum in the relevant limits of the scalar couplings, according to the discussion on the accidental global symmetries in Sect. 3.3. D.2.1 45 only

′ Z′ Applyingω ωR theωB stationaryL conditions in Eqs. (3.12)–(3.13), to the ﬂipped vacuum with − − , we ﬁnd 5 1 M2 ; a ω2 ; = = − 2 M2 ; ; (24 0)− = 4 (D.12) M2 ; a a ω2 ; (10 4)=0 1 2 (1 0)=2 (80 + 13 ) aand, as expected, the spectrum exhibits 20 WGB and 24 PGB whose mass depends on 2 only. The required positivity of the scalar masses gives the constraints a < a > 13 a ; 2 and 1 − 80 2 (D.13) where the second equation coincides0 with the constraint coming from the stability of the scalar potential (see Eq. (C.5) in Appendix C). D.2.2 16 only

H When only the part of the scalarSU potential is considered the symmetry is spontaneously broken to the standard gauge group. Applying the stationary Eq. (3.14) 16 we ﬁnd (5) M2 λ χR2 ; 2 M2 ; (5)=2 (D.14)

M2(10) = 0 λ χR2 ; ! 1 1 1 1 (1) = 1 1 2 Appendix D. Tree level mass spectra 176

ψ ψ SU 16 16∗ in the ( 1 , 1 ) basis, with the subscripts referring to the standard origin, that yields (5)

M2 ; Det M2 λ χR2 ; Tr (1) =1 0 (D.15) (1) = λ and as expected we count 21 WGB and 10 PGB modes whose mass depends on 2 only. The required positivity of the scalar masses leads to

λ > λ > ; 2 and 1 (D.16) 0 0 where the second equation coincides with the constraint coming from the stability of the scalar potential (see Eq. (C.3) in Appendix C).

D.2.3 Mixed 45-16 spectrum (χR ) 6

In the general case the unbroken symmetry= 0 is the SM group. Applying ﬁrst the two stationary conditions in Eq. (3.12) and Eq. (3.14) we ﬁnd the spectrum below. The × ψ45 ψ16 matrices are spannedψ ψ ψ overψ the ( , ) basis whereas the × SM singlet matrix is 45 45 16 16∗ 2 2 given in the ( 15, 1 , 1 , 1 ) basis. 4 4

βχR2 a ωB L ωR ωB L χR τ βωB L M2 ; ; 2 − − − − ; χR τ βωB L ωR τ βωB L ! − − − − +2 ( + ) ( 3 ) (1 1 +1) = βχR2 a ωR ωR ωB L χR τ β ωR ωB L M2 ; ; 2 (2 3 ) − 2 (− 3 ) − ; − 3 χR τ β ωR ωB L ωB L τ β ωR ωB L ! (D.17) − − − − − +2 ( + ) ( (2 + )) (3 1 ) = M2 ; ; a ωB L ωR ωB L ωR ; 2 − (− (2 − + )) 2 ( (2 + )) M2 ; ; a ωR ωB L ωR ωB L ; (1 3 0)=2 2( − − )( +2 − ) (D.18) M2 ; ; 5 ; (8 1 −0)=26 ( )( +2 ) βχ a ω ω χ τ β ω ω (3 2 )=0 R2 R B L R R B L M2 ; ; 1 2 − − − ; 6 χR τ β ωR ωB L ωR ωB L τ β ωR ωB L ! − − − − − +4 ( ( +2 )) M2(3; 2; + 1) = ωR ωB L τ βωR λ χR2 ; − 2 ( − ( +2− )) (2 + ) ( ( +2 )) (D.19) M2 ; ; 1 ωR ωB L τ βωB L λ χR2 : (1 2 3) = ( +3 − ) ( − )−+2 2 (3 1 + )=2 ( + ) ( ) +2 D.2. Anatomy of the scalar spectrum 177

1 βχR2 a ωR2 a ωB LωR a a ωB2 L 2 2 2 − 1 2 − βχR  √ a a ωRωB L M2 ; ; 2 1 2 − 3 +4√ 2 + + (48 +7 )  1 χR τ βωR α β ωB L  − 2 6 +− 2(16 −+3 ) −  1 √ χR τ βωR α β ωB L (1 1 0) =  − − − −  2 3 ( 2 (16 +3 ) ) βχR √ a a ωRωB L 1 √ χR τ βωR α β ωB L 2 3 ( − 2 (16 +3 ) ) − 2 1 2 − 2 − − χR τ α β ωR βωB L βχR2 a a ωR2 a ωB LωR a ωB2 L (− +2(8 +√ ) +3 − ) 1 2 2 − 2 − 6 χR+τ 2(16α β ω+3R βωB) L 3 ( 2 2 (16 +3 ) ) (− +2(8 +√ ) +3 − ) 1 λ χR2 1 +2 4(8 + ) 2+ + χR τ 2α β ωR βωB L 1 λ χR2 (− +2(8 +√ ) +3 − ) 2 1 2

1 √ χR τ βωR α β ωB L − − 2 χR −τ α β−ωR βωB L − − ( +2(8 +√ ) +3 )  : 3 ( 2 λ 2χ (16 +3 ) ) 1 R2  (D.20) 2 1  1 λ χR2  1  2  By applying the remaining stationary condition in Eq. (3.13) one obtains

M2 ; ; ; Det χR2 ωR2 τ βωB L M ; ; − − ; 2 (1 1 +1) = 0 ω Tr R M ; ; ; +4 ( 3 ) (12 1 +1)− 2 = Det 3 2 χR2 ωB2 L τ β ωR ωB L M2 (;3 ;1 2 )=0 − − − ; − 3 ωB L Tr − (D.21) M ; ; ; +4 ( (2 + )) (23 1 )1 = Det 6 2 M2 ; ; 1 βχR2 a ωRωB L ωR ωB L τ β ωR ωB L ; − − − Tr (3 2 +6 )=0 2 − M2 ; ; ; Rank (3 2 + ) = +4 + ( + ) ( ( +2 )) M2 ; ; a a ωR2 a a ωB2 L a ωRωB L χR2 5 β λ : Tr (1 1 0)=3 1 2 1 2 − 2 − 2 1 (1 1 0)=2 (32 +5 ) + 8(6 + ) +2 + + SOIn Eqs./SM (D.17)–(D.21) we recognize the 33 WGB with the quantum numbers of the coset algebra. In using the stationary condition in Eq. (3.13), we paid attention not to divide by ωR(10)ωB L ω ωR ωB L ( − ), since the ﬂipped vacuum − − is an allowed conﬁguration. ωR ωB L On the other hand, we can freely put and − into the denominators, as the vacua ωR + ωB L = = a and − are excluded at the tree level. The coupling 2 in Eq. (D.21) is understood to obey the constraint = 0 = 0 a ωR ωB L ωRωB L βχR2 ωR ωB L τχR2 : 2 − − − − (D.22) 4 ( + ) + (2 +3 ) =0 Appendix D. Tree level mass spectra 178 D.2.4 A trivial 45-16 potential (a λ β τ )

It is interesting to study the global symmetries2 2 of the scalar potential when only the H H = = = = 0 moduli of and appear; ; in the scalar potential. In ordera toλ correctlyβ τ count the corresponding PGB, the massψ ψ matrixψ inψ the limit of 2 2 needs 45 16 45 45 16 16∗ to be scrutinized. We ﬁnd in the ( 15, 1 , 1 , 1 ) basis, (1 1 0) = = = =0 M2 ; ;

a ωB2 L √ a ωRωB L √ αχRωB L √ αχRωB L (1 1 0) = 1 − 1 − − − √ a ωRωB L a ωR2 √ αχRωR √ αχRωR  −  ; 96 1 32 6 1 8 3 8 3 √ αχRωB L √ αχRωR 1 λ χR2 1 λ χR2 (D.23) −  32 6 64 8 22 1 8 22 1   √ αχRωB L √ αχRωR 1 λ χR2 1 λ χR2   −   8 3 8 2 2 1 2 1  8 3 8 2 with the properties

M2 ; ; ; Rank M2 ; ; a ωR2 a ωB2 L λ χR2 : Tr (1 1 0)=21 1 − 1 (D.24) (1 1 0) = 64 + 96 + a λ β Asτ expected from the discussion in Sect. 3.3, Eqs. (D.17)–(D.23) in the 2 2 limit exhibit 75 massless modes out of which 42 are PGB. = = = D.2.5=0 A trivial 45-16 interaction (β τ ) α In this limit, the interaction part of the potential consists only of the term, which H H = = 0 is the product of and moduli. Once again, in order to correctly count the ; ; β τ ψ45 ψ45 masslessψ ψ modes we specialize the matrix to the limit. In the ( 15, 1 , 16 16∗ 45 16 1 , 1 ) basis, we ﬁnd (1 1 0) = =0 M2 ; ;

a ωR2 a ωB LωR a a ωB2 L √ a a ωRωB L (1 12 0) = 2 − 1 2 − 1 2 − √ a a ωRωB L a a ωR2 a ωB LωR a ωY2  1 2 − 1 2 2 − 2 2 + √ αχ+Rω (48B L +7 ) 2 6(16 √ +3αχRω)R  −  2 6(16√ αχ+3RωB) L 2 4(8 + ) √ +αχRωR +  −  8 3 8 2 √ αχRωB L √ αχRωB L − − 8 3 √ αχ ω 8 2√ αχ ω R R R R  ; 8 3 8 3 1 λ χR2 1 λ χR2 (D.25) 1 1  8 22λ χ 8 22λ χ  1 R2 1 R2  2 1 2 1  D.2. Anatomy of the scalar spectrum 179 with the properties

M2 ; ; ; Rank M2 ; ; a a ωR2 a a ωB2 L a ωRωB L λ χR2 : Tr (1 1 0)=3 1 2 1 2 − 2 − 1 (D.26) (1 1 0)=2 (32 +5 ) + 8(6 + ) +2 + β Accordingτ to the discussion in Sect. 3.3, upon inspecting Eqs. (D.17)–(D.21) in the limit, one ﬁnds 41 massless scalar modes of which 8 are PGB. = =0 D.2.6 The 45-16 scalar spectrum for χR

χR The application of the stationary conditions in Eqs. (3.12)= 0–(3.13) (for , Eq. (3.14) is trivially satisfied) leads to four different spectra according to the four vacua: standard Z ′ Z′ C L R B L C L R =0 , flipped , − and . We specialize our discussion to the last three cases. 51 5 1 3 2 2 1 4 2 1 SOThe mass eigenstates are conveniently labeled according to the subalgebras of left invariant by each vacuum. With the help of Tables 4.4–4.χR5 one can easily recover the decomposition in the SM components. In the limit the states H(10) H H χR H and do not mix. All of the WGB belong to the , since for the SO = 0 preserves . ω ω ω 45 16 R B L 45 =0′ Z′ 16 Consider first the case: − − (which preserves the flipped group). H (10) For the components we obtain: = = 5 1 M2 ; a ω2 ; 45 − 2 M2 ; ; (24 0)− = 4 (D.27) M2 ; a a ω2 : (10 4)=0 1 2

H Analogously, for the components(1 0)=2 we(80 get: + 13 ) M162 ; 1 ω2 α β τω ν2 ; 4 − M2 ; 1 ω 2 α β τω ν2 ; (10−+1) =4 (80 + )+2− − 2 (D.28) M2 ; 1 ω2 α β τω ν2 : (5¯ 3) = 4 (80 +9 ) 6 2− (1 +5) = 5 (16 ′+5Z′ )+10 2 Since the unbroken groupH is the ﬂipped we recognize, as expected, 45-25=20 WGB. When only trivial invariants (moduli) are considered the global symmetry O 5 1 ω O of the scalar potential is , broken spontaneously by to . This leads to 44 45 GB in the scalar spectrum. Therefore 44-20=24 PGB are left in the spectrum. On (45) (44) general grounds,a theirβ massesτ should receive contributions from all of the explicitlya breaking terms 2, and . As it isM directly; seen from thea spectrum,< only the 2 term contributes at the tree level to . By choosing 2 one may obtain a (24 0) 0 Appendix D. Tree level mass spectra 180 consistent minimum of the scalar potential. Quantum corrections are not relevant in this case. ωR ωB L C L R B L − 6 − Consider thenH the case and which preserves the gauge group. For the components we obtain: =0 =0 3 2 2 1 45 M2 ; ; ; a ωB2 L ; 2 − M2 ; ; ; a ωB2 L ; (1 3 1 0)=2 2 − M2 ; ; ; a ωB2 L ; (1 1 3 0)=2− 2 − M2 ; ; ; 1 ; (8 1 1 −0)3 = 4 (D.29) M2 ; ; ; 2 ; (3 2 2 − 3)=0 M2 ; ; ; a a ωB2 L : (3 1 1 )=0 1 2 − (1 1 1 0)=2 (48 +7 ) H Analogously, for the components we get: 16 M2 ; ; ; 1 1 ωB2 L α β τωB L ν2 ; − 6 4 − − − M2 ; ; ; 1 1 ωB2 L α β τωB L ν2 ; (3 2 1 −+ 6) = 4 − (48 + ) 2 − − 2 M2 ; ; ; 1 1 ωB2 L α β τωB L ν2 ; (3 1 2 − 2) = 4 − (48 + )+2 − −2 M2 ; ; ; 1 1 ωB2 L α β τωB L ν2 : − (1 2 1 2) = 4 − (48 +9 )+6− − 2 (D.30) (1 1 2 + ) = (48 +9 ) 6 2 ; ; ; ; ; ; Worth of a note is the mass degeneracy of the and multiplets which ωR ωB L is due to the fact that for D-parity is conserved by even − powers. On the H (1 3 1 0) (1 1τ 3 0) contrary, in the components the D-parity is broken by the term that is linear in ωB L = 0 − . 16 C L R B L Since the unbroken group is − there are 45-15=30 WGB, as it appears from the explicit pattern of the scalar spectrum. When only trivial invariants (moduli H 3 2 2 1 O terms) of are consideredO the global symmetry of the scalar potential is , broken spontaneously to , thus leading to 44 GB in the scalar spectrum. As 45 (45) a consequence 44-30=14 PGB are left in the spectrum. On general grounds, their (44) a β τmasses should receive contributions from all of the explicia tly breaking terms 2, and . As it is directly seen from the spectrum, only the 2 term contributes at the tree level to the mass of the 14 PGB, leading unavoidably to a tachyonic spectrum. This feature is naturally lifted at the quantum level. ωR ωB L C L R Let us ﬁnally consider the case 6 and − (which preserves the =0 =0 4 2 1 D.2. Anatomy of the scalar spectrum 181

H gauge symmetry). For the components we ﬁnd:

M2 ; ; a ωR2 ; 45 2 M2 ; ; a ωR2 ; (15 1 0)=2− 2 M2 ; ; 1 ; (1 3 0)2 = 4 (D.31) M2 ; ; 1 ; (6 2 −+ 2)=0 M2 ; ; ; (6 2 )=0 M2 ; ; a a ωR2 : (1 1 +1) = 0 1 2 H For the components we obtain:(1 1 0)=8 (8 + ) M ; ; αω ν ; 16 2 R2 1 2 − 2 M2 ; ; 1 ωR2 α β τωR 1 ν2 ; (4 2 0)=82 − 2 (D.32) M2 ; ; 1 ωR2 α β τωR 1 ν2 : (4 1 +− 2) = (8 + )− + − 2 (4 1 ) = (8 + ) C L R The unbroken gauge symmetry in this case corresponds to . Therefore, oneH can recognize 45-19=26 WGB in the scalar spectrum. When only trivial (moduli) 4 2 1 O invariants are considered the global symmetry of the scalar potential is , which is ωR O 45 broken spontaneously by to . This leads globally to 44 massless states in the H (45) scalar spectrum. As a consequence, 44-26=18 PGB are left in the spectrum, that (44) a β τ should receive mass contributions from the explicitly breaking terms 2, and . At a 45 the tree level only the 2 term is present, leading again to a tachyonic spectrum. This is an accidental tree level feature that is naturally lifted at the quantum level. Appendix D. Tree level mass spectra 182 Appendix E

One-loop mass spectra

We have checkedχR explicitly that the one-loop corrected stationaryωR equationωB L ω (3.13)R main-ωB L tains in the limit the four tree level solutions, namely, − , − − , ωR ωB L Z ′ Z and − , corresponding respectively to the standard , ﬂipped ′ , C L R B L =0C L R = = − and vacua. = 0 = 0 51 5 1 3 2In2 what1 follows4 2 we1 list, for the last three cases, the leading one-loop corrections, arising from the gauge and scalar sectors, to the critical PGB masses. For all other states the loop corrections provide only sub-leading perturbations of the tree-level masses, and as such irrelevant to the present discussion.

E.1 Gauge contributions to the PGB mass

Before focusing to the three relevant; ; vacuum; ; conﬁgurations, it is convenient to write the gauge contribution to the and states in the general case. (1 3 0) (8 1 0) g 4 ωR2 ωB LωR ωB2 L M ; ; − − 2 π 2 16 g 4+ + 19 g 2 ωR ωB L 2 ∆ (1 3 0) = ωR ωB L 3 − − π ωR ωB4L − − µ 2 − − 2 3 ( ) + 2 ( ) logg 2 ωR ωB L 2 4 (ωR ωB L) ωR ωB L 2 − − − − µ 2 ( + ) + (4 5 ) ( + ) glog2ωR2 g 2ωB2 L ωR3 ωB3 L − ; − µ − µ 2 2 (E.1) 4 4 4 log +8 log Appendix E. One-loop mass spectra 184

g 4 ωR2 ωB LωR ωB2 L M ; ; − − 2 π 2 13 + + 22 ∆ (8 1 0) = g 4 g 2 ωR ωB L 2 ωR ωB L 3 − − π ωR ωB4 L − − µ 2 − − 2 3 ( ) + ( ) log g 2 ωR ωB L 2 8 ω(R ωB L ) ωR ωB L 2 − − − − µ 2 ( + ) + (5 7 ) ( + ) logg 2ωB2 L g 2ωR2 ωY2 ωR ωB L − ωR3 : − µ − µ 2 2 (E.2) 4 4 +4 (3 + ) log 8 log One can easily recognize the (tree-level) masses of the gauge bosons in the log’s arguments and cofactors (see Appendix F.3.2). Note that only the massive states do contribute to the one-loop correction. (see Sect. 3.4.3). ′ Z′ ω Let’sωR nowωB specializeL to the three relevant vacua. First, for the ﬂipped case − − one has: g ω g ω g ω 5 1 = = M ; 4 2 4 2 2 2 : 2 π π µ 2 2 2 (E.3) 17 3 4 ωR ∆ (24ωB L0) = C L +R B L log Similarly, for and − 6 2( 2− ): =0 =0 3 2 2 1 M2 ; ; ; M2 ; ; ;

g 4ωB2 L g 4ωB2 L g 2ωB2 L g 4ωB2 L g 2ωB2 L − − − − − ; ∆ (1 3 1 0)=∆ π(1 1 3 0) π µ − π µ 2 2 2 2 2 19 21 24 4 = g 4ωB2 L + g 4ωB2 L log g 2ωB2 L log M2 ; ; ; 4 − 4 − − : 4 π π µ 2 2 2 (E.4) 11 3 ∆ (8 1 1ωR0) = ωB L + C LlogR Finally, for 6 and2 − 2( ): 4

=0 g ω =0g ω 4 2 1 g ω M ; ; 4 R2 4 R2 2 R2 ; 2 π π µ 2 2 2 4 3 ∆ (1 3 0) = g 4ω+R2 g 4ωlogR2 g 2ωR2 g 4ωR2 g 2ωR2 M2 ; ; 2 16 : π π µ − π µ 2 2 2 2 2 (E.5) 13 9 12 4 ∆ (15 1 0) = + log log E.2 Scalar contributions4 4 to the PGB4 mass

Since the general formula for the SM vacuum conﬁguration is quite involved, we give directly the corrections to the PGB masses on the three vacua of our interest. We E.2. Scalar contributions to the PGB mass 185

ω ωR ωB L ′ Z consider ﬁrst the case − − (ﬂipped ′ ): = = 5 1

τ2 β2ω2 M2 ; π2 + 5 ω α β τω ν ∆ (24 0) = βω τ ω αω βω τ ν 2 − 2 (E.6) π ω − 4 − − 2 µ 2 2 1 5 (16 + 5 ) + 10 2 + ω τω ( α5 β )(5βω(162 β + 5 α+ 2 ) τ22 )ν log2 βω τ 128 − − − 4 ω α β τω ν + 3 2 (80 + 3 )+− (27− 2 400 ) 10 + (10 6 ) × µ 2 (80 + 9 ) 6 2 logω τ βω αω βω2 α β τ2 ν2 τ βω − 4 − ω α β τω ν + 2 2(33 80 )+− (4002 : + 17 ) + 10 + 2 ( 5 ) × µ 2 (80 + ) + 2 2 log 4

ωR ωB L C L R B L For and − 6 ( − ), we ﬁnd: =0 =0 3 2 2 1

τ2 β2ωB2 L M2 ; ; ; M2 ; ; ; − π2 + 2 ∆ (1 3 1 0) = ∆τ (1βω1B 3L 0) = ωB2 L α β τωB L ν2 (E.7) π ωB L − − − − − 4 − 2 − h 1ω α β τω ν + B2 L ( 3 ) B L3 2(16 + 3 ) + 6 + 2 64 − − − − × µ2 ! (48 + 9 ) 6 2 log ωB2 L α β τωB L ν2 βωB L τ ωB2 L 4 α β τωB L ν2 − − − − − − − − µ2 ! (48 + ) + 2 2 ( τωB2 +L )α β(48 βω+ B3) +L 2 α β2 logωB L τ2 βν2 ν2τ − − − − − 4− ωB2 L α β τωB L ν2 + 3 −(16 11 )+− − (240− + 17 ) + 2 5 5 2 × µ2 ! (48 + ) 2 2 logωB2 L βτ ατ βωB3 L β α ωB L βν2 τ2 ν2τ − − 4 − − − − ωB2 L α β τωB L ν2 + (9− 48 ) + 3 −(9− 16 ) +; 2 + 2 × µ2 !# (48 + 9 ) + 6 2 log 4 Appendix E. One-loop mass spectra 186

τ2 β2ωB2 L M2 ; ; ; − π2 + 3 ∆ (8 1 1 0) = τ βωB L ωB2 L α β τωB L ν2 (E.8) π ωB L − − 4 − − − − 2 − h 1ω α β τω ν + B2 L ( 3 ) B L3 2(16 + 3 ) + 6 + 2 64 − − − − × µ2 ! (48 + 9 ) 6 2 logωB2 L βτ ατ βωB3 L α β ωB L τ2 βν2 ν2τ − − 4 − − − ωB2 L α β τωB L ν2 + (21− 48 )+ − (144− + 11 )+ 6 6 + 2 × µ2 ! (48 + ) + 2 2 log ωB2 L α β τωB L ν2 βωB L τ ωB2 L4 α β τωB L ν2 − − − − − − − − µ2 ! (48 + 9 ) + 6 2 (3 τωB2 L+ α) β (48βω+B3 9L ) +α 6 β 2ωB Llog τ2 βν2 ν2τ − − − − − − 4 ωB2 L α β τωB L ν2 + 3 −(16 7 )+− −(144− + 11 :)+ 6 6 2 × µ2 !# (48 + ) 2 2 log ωR 4 ωB L C L R Finally, for 6 and − ( ), we have: τ2 β2ωR2 M2 ; ; =0 =0 4 2 1 π2 + 2 αω ν ∆ (1 3 0) = ω αβω βν τ R2 − (E.9) R 4 R2 2 2 22 π2ωR " − µ2 ! 1 8 ν + 16 16 + log ωR2 α β τωR 64τ βωR ωR2 α β τωR ν2 − − 22 − − − µ2 ! (8 + ) ν 4 ( 2 ) 2 (8 + ) + 2 + log ωR2 α β τωR βωR τ ωR2 α β τωR ν2 − 22 ; − − µ2 !# (8 + )+ 4 (2 + ) 2τ2 (8β2ω+R2 ) + 2 log M2 ; ; 4 π2 + αω ν ∆ (15 1 0) =ω αβω βν τ R2 − (E.10) R 4 R2 2 2 22 π2ωR " − µ2 ! 1 8 ν + 8 16 + log ωR2 α β τωR 64 βωR3 α β ατωR2 ωR τ2 βν2 ν2τ − − 22 − − − − µ2 ! (8 + ) ν 4 2 (8 ) 16 + + log ωR2 α β τωR βωR3 β α ατωR2 ωR βν2 τ2 ν2τ − 22 : − − − µ2 !# (8 + )+ +4 2 ( 8 ) 16 + + log H Also in these formulae we recognize the (tree level) mass eigenvalues of the states contributing to the one-loop eﬀective potential (see Appendix D.2.6). 16 E.2. Scalar contributions to the PGB mass 187

; ; ; ; ; Notice; that the singlets′ Z withC L respectR B L to eachC L vacuum,R namely , and , for the ﬂipped ′ , − and vacua respectively, receive a tree a a (1 0) a(1 1 1 0) level contribution from both 1 as well as 2 (see Appendix D.2.6). The 1 term leads (1 1 0) 5 1 3 2 2 1 4 2 1 the tree level mass and radiative corrections can be neglected. One may verify that in the limit of vanishing VEVs the one-loop masses vanish identically on each of the three vacua, as it should be. This is a non trivial check of the calculation of the scalar induced corrections. Appendix E. One-loop mass spectra 188 Appendix F

Flipped SO vacuum

F.1 Flipped SO(10)notation

We work in the basis of Ref. [223], where the adjoint is projected along the positive- chirality spinorial generators(10) ij ij+ ; ≡ (F.1) i;j ;::; with . Here 45 45 Σ + I χ ; =1 10 − ! ≡ 32 ± (F.2) Σ 1 I ( Γ ) Σχ where 32 is theγ -dimensional identityΣ matrix2 and is the 10-dimensional analogue of the Dirac 5 matrix defined as 32 Γ χ i : ≡ − 1 2 3 4 5 6 7 8 9 10 (F.3) i Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ σThei factors are given by the followingI tensor products of ordinary Pauli matrices and the -dimensional identity 2: Γ σ σ I I σ ; 2 1 ≡ 1 ⊗ 1 ⊗ 2 ⊗ 2 ⊗ 2 σ σ I σ σ ; 2 ≡ 1 ⊗ 2 ⊗ 2 ⊗ 3 ⊗ 2 Γ σ σ I σ σ ; 3 ≡ 1 ⊗ 1 ⊗ 2 ⊗ 2 ⊗ 3 Γ σ σ I σ I ; 4 ≡ 1 ⊗ 2 ⊗ 2 ⊗ 2 ⊗ 2 Γ σ σ I σ σ ; 5 ≡ 1 ⊗ 1 ⊗ 2 ⊗ 2 ⊗ 1 Γ σ σ I σ σ ; 6 ≡ 1 ⊗ 2 ⊗ 2 ⊗ 1 ⊗ 2 Γ σ σ σ I I ; 7 ≡ 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 2 Γ σ σ σ I I ; 8 ≡ 1 ⊗ 3 ⊗ 2 ⊗ 2 ⊗ 2 Γ σ σ σ I I ; 9 ≡ 1 ⊗ 3 ⊗ 3 ⊗ 2 ⊗ 2 Γ σ I I I I ; 10 ≡ 2 ⊗ 2 ⊗ 2 ⊗ 2 ⊗ 2 Γ (F.4) Γ Appendix F. Flipped SO vacuum 190 (10) which satisfy the Clifford algebra i; j δij I : { } 32 (F.5) ij The spinorial generators, , are thenΓ Γ defined=2 as i Σ ij i; j : ≡ (F.6) SO Σ Γ Γ On the flipped vacuum the adjoint4 representation reads (10) h iL · ; h i (F.7) · h iR ! 45 45 = where 45 L λ ; λ ; λ ; λ ; λ ; λ ; λ ; λ ; h i diag 1 2 3 4 5 6 7 8 (F.8) and 45 λ= ( ω+ ) 9 · · · · · · λ ω+  · 10 · · · · ·  λ ω+  · · 11 · · · ·   λ ω   +  R  · · · 12 · · ·  : h i  ω− λ  (F.9)  · · · 13 · · ·   ω λ  45 =  −   · · · · 14 · ·   ω− λ   · · · · · 15 ·   ω− λ   · · · · · · 16    In the convention defined in section 4.4.2 (cf. also caption of Table 4.5), the diagonal entries are given by ωB L λ λ λ λ λ λ − ; 1 2 3 5 6 7 √ (F.10) ωB L λ = λ = = − =; = = 4 8 − √ 2 2 3 ωB L ωR ωB L ωR λ = λ = λ − ; λ − ; 9 10 112 2− √ − √ 12 √ − √ ωB L ωR 3ωB L ωR λ = λ = λ = − ; λ = − : 13 14 15 − 2√2 √2 16 2√2 √2 3 ωB L ωR = = = ω+ ω+−∗ = + where − and are real, while 2 2 . 2 2 2 2 Analogously, the spinor and the anti-spinor SM-preserving vacuum directions are = given by T e ν ; h i T ··········· ··· − (F.11) ν e ; 16 = ( ··· ··· ········ ) (F.12)

16 = ( ) F.2. Supersymmetric vacuum manifold 191 where the dots stand for zeros, and the non-vanishing VEVs are generally complex. It is worthT remindingT thatT the shorthand notation and in Eq. (4.19) standsT for C and C , where C is the “charge conjugation” matrix obeying + − 16 16 1645 16 C C . In the current convention, C is given by 16 16 16 45 16 (Σ ) + Σ =0 I · · ·− 4 I  · · 4 ·  ; C I (F.13)  · 4 · ·   I  =    − 4 · · ·  I where 4 is the four-dimensional identity matrix.

F.2 Supersymmetric vacuum manifold

MU In order for SUSY to surviveD the spontaneousF GUT symmetry breakdown at the WvacuumH manifold must be - and -ﬂat at the GUT scale. The relevant superpotential given in Eq. (4.19), with the SM-preservingF vacuum parametrized by Eq. (F.7) and Eqs. (F.11)–(F.12), yields the following -ﬂatness equations: τ τ FωR µωR 1 e e ν ν 2 e e ν ν ; − √ 1 1 − 1 1 √ 2 2 − 2 2 τ τ 2 FωB L µωB L 1 e e ν ν 2 e e ν ν ; − = 4 − + ( ) + ( )=0 3 √2 1 1 1 1 √2 2 2 2 2 Fω µω− τ ν e τ ν e ; =4 −+1 1 1( − 2+2 2 ) + ( + )=0 + 2 2 Fω− µω+ τ e ν τ e ν ; =4 − 1 1 1 − 2 2 2 =0 =4 =0

e ωR e ωB L Fe τ ω−ν 1 1 − ρ e ρ e ; 1 − 1 − √ √ 11 1 12 2 1 e ωR 3e ωB L Fe = τ ω−ν 2 + 2 − + ρ e + ρ e =0 ; 2 − 2 − √2 2√2 21 1 22 2 2 ν ωR 3ν ωB L Fν = τ ω+e 1 + 1 − + ρ ν + ρ ν =0; 1 − 1 √2 2√2 11 1 12 2 1 ν ωR 3ν ωB L Fν = τ ω+e + 2 + 2 − + ρ ν + ρ ν =0 ; 2 − 2 √2 2√2 21 1 22 2 2 3 = + + + + =0 2 2 2 Appendix F. Flipped SO vacuum 192

e ωR e ωB L (10) Fe τ ω+ν 1 1 − ρ e ρ e ; 1 − 1 − √ √ 11 1 21 2 1 3 = e ωR + e ωB L + + =0 Fe τ ω+ν 2 2 22 2− ρ e ρ e ; 2 − 2 − √ √ 12 1 22 2 2 3 = ν ωR + ν ωB L + + =0 Fν τ ω−e 1 2 21 2− ρ ν ρ ν ; 1 − 1 √ √ 11 1 21 2 1 3 = + ν ωR + ν ωB L + + =0 Fν τ ω−e 2 2 22 2− ρ ν ρ ν : 2 − 2 √ √ 12 1 22 2 (F.14) 2 3 = + + + ωR+ ωB L =0ω+ ω− One can use the ﬁrst four equations2 above2 to2 replace , − , and in the remaining eight (complex) relations which can be rewritten in the form ω µFe µ ρ e ρ e 11 1 12 2 1 τ2 ν ν e e e τ τ ν ν e ν ν e e e ; 16 = 16 ( + − 1) 1 1 1 1 1 − 1 2 2 2 1 2 1 2 1 2

ω 5 ( + ) ( + (4 +5 ) ) =0 µFe µ ρ e ρ e 11 1 21 2 1 τ2 ν ν e e e τ τ ν ν e ν ν e e e ; 16 = 16 ( + − )1 1 1 1 1 1 − 1 2 2 2 1 2 1 2 1 2

ω 5 ( + ) ( + (4 +5 ) ) =0 µFν µ ρ ν ρ ν 11 1 12 2 1 τ2 e e ν ν ν τ τ e e ν e e ν ν ν ; 16 = 16 ( + − )1 1 1 1 1 1 − 1 2 2 2 1 2 1 2 1 2

ω 5 ( + ) ( + (4 +5 ) ) =0 µFν µ ρ ν ρ ν 11 1 21 2 1 τ2 e e ν ν ν τ τ e e ν e e ν ν ν ; 16 = 16 ( − + 1 1 )1 1 1 1 − 1 2 2 2 1 2 1 2 1 2 (F.15) 5 ( + ) ( + (4 +5 ) ) =0 where the other four equationsD are obtained from these by exchanging ↔ . There are two classes of -ﬂatness conditions corresponding, respectively, to the U X SO X 1 2 VEVs of the and the generators. For the -charge one ﬁnds

DX † X(1) † X (10) † X † X † X h i h i h 1i h 1i 1 1 h 2i h 2i 2 2 e 2 ν 2 e 2 ν 2 e 2 ν 2 e 2 ν 2 ; = 45 45 +| 161| | 1|16−|+1| 16−| 1| 16| 2|+ 16| 2| −|162| −|+ 162| 16(F.16) SO while for the = generators+ one has + + =0

Dij Dij45 Dij16⊕16 ; (10) ≡ (F.17) where + =0 Dij45 † ij+; ; Tr h i h i (F.18) = 45 Σ 45 F.2. Supersymmetric vacuum manifold 193 and † † Dij16⊕16 † ij+ ij− † ij+ ij− : h 1i h 1i 1 1 h 2i h 2i 2 2 (F.19)

Given that= 16 Σ 16 + 16 Σ 16 + 16 Σ 16 + 16 Σ 16 † ij+; ij+ ; † ; Tr h i h i Tr h i h i (F.20) h i we obtain 45 Σ 45 = Σ 45 45 ; † · · ; h i h i DR ! (F.21) h i · 45 45 = where A √ B∗ · · · · · · A √ B∗  · · · · · ·  A 2 √ B∗  · · · · · ·   A 2 √ B   ∗  DR  · · · · · ·  ;  √ B A 2  (F.22)  · · ·− · · ·   √ B A 2  =    · · · · − · ·   2 √ B A   · · · · ·− ·   2 √ B A   · · · · · ·−   2  and 2 A ω 2 ω− 2 ; | +| −| | B ω ∗ ωR ωR ∗ ω− : + − = (F.23) ωR ω+ ω− ∗ Dij45 Fω± Since is real and , = ( as) it should( ) be. Notice that -ﬂatness implies τ e ν τ e ν τ ν e ∗ τ ν e ∗ =1 ( 1 )1 2 =02 2 1 1 1 2 2 2 (F.24) τ ; where the reality of 1 2 has been+ taken= into( account.) + ( ) For the spinorial contribution in (F.17) we ﬁnd

Dij16⊕16 ij+ ; e 2 e 2 ij+ ; ν 2 ν 2 12 12 | 1| | 2| 16 16 | 1| | 2| ij− ; ν 2 ν 2 ij− ; e 2 e 2 = (Σ ) + 4 4 | 1+| (Σ |) 2| +8 8 | 1| | 2| ij+ ; e∗ν e∗ν ij+ ; ν∗e ν∗e −+ (Σ )12 16 1 1+ 2 2 +− (Σ )16 12 1 1+ 2 2 ij− ; ν∗e ν∗e ij− ; e∗ν e∗ν : 4 8 1 1 2 2 8 4 1 1 2 2 (F.25) T(Σ ) ( + ) (Σ ) ( + ) − −1 + Given −C C and the explicit+ (Σ ) form( of+C in Eq.) + (F.13), (Σ ) one( can+ verify) readily that Σ = (Σ ) ij− ; ij+ ; ; 4 4 − 16 16 ij− ; ij+ ; ; 8 8 − 12 12 (Σ ) = (Σ ) ij− ; ij+ ; : (Σ )4 8 = (Σ )12 16 (F.26) (Σ ) = +(Σ ) Appendix F. Flipped SO vacuum 194 (10) Dij16⊕16 Thus, can be simpliﬁed to

ij+ ; e 2 e 2 e 2 e 2 ij+ ; ν 2 ν 2 ν 2 ν 2 12 12 | 1| | 2| −| 1| −| 2| 16 16 | 1| | 2| −| 1| −| 2| ij+ ; e∗ν e∗ν ν∗e ν∗e ; (Σ ) ( + − )+(Σ12 16 )1 1 ( 2 2+− 1 1 − 2 2 c.c.) (F.27) or, with Eq. (F.16) at hand, to (Σ ) ( + ) + =0

ij+ ; ij+ ; ν 2 ν 2 ν 2 ν 2 16 16 − 12 12 | 1| | 2| −| 1| −| 2| ij+ ; e∗ν e∗ν ν∗e ν∗e : (Σ ) (Σ ) ( +− 12 16 1 1 ) 2 2 − 1 1 − 2 2 c.c. (F.28) ij+ Taking into account the basic(Σ features) ( of+ the spinorial generators) + =0(e.g. , the ij+ ; ij+ ; ij+ ; bracket [ 16 16 − 12 12 and 12 16 can never act against each other because at ij+ ; Σ least one of them always vanishes, or the fact that 12 16 is complex) Eq. (F.28) can (Σ ) ij(Σ ) ] (Σ ) be satisfied for all if and only if (Σ ) e 2 e 2 e 2 e 2 ; | 1| | 2| −| 1| −| 2| ν 2 ν 2 ν 2 ν 2 ; | 1| | 2| −| 1| −| 2| e∗ν + e∗ν ν∗e ν∗e = 0 ; 1 2 − 1 − 2 1 + 2 1 2 = 0 (F.29) D F Combining this with Eq. (F.24), the+ required - and -flatness= 0 can be in general main- e∗; e ; ν∗; ν ; tained only if 1 2 1 2 and 1 2 1 2. Hence, we can write iφe iφe = e ; e=; e ; ; e ; e ; e− ; ; 1 2 ≡| 1 2| 1 2 ≡| 1 2| iφν 1 2 iφν 1 2 ν ; ν ; e ; ; ν ; ν ; e− ; : 1 2 ≡| 1 2| 1 2 ≡| 1 2| 1 2 1 2 (F.30) F With this at hand, one can further simplify the -flatness conditions Eq. (F.15). To this end, it is convenient to define the following linear combinations V V LV− C φV C φV ; ≡ 1V − V2 (F.31) L C φV C φV ; V+ ≡ 1 cos 2 sin (F.32) where sin + cos V ω ω V ω ω C FV FV ; C FV FV ; 1 ≡ i − 2 ≡ 1 1 V e e ν +ν µ τ τ with running over the2 spinorial VEVs 1, 2, 21 and 2. For , 1 and 2 real by LV± V definition, the requirement of for all is equivalent to

µ Le− e τ τ ν ν =0φe φe φν φν Re | 2| 1 2 | 1|| 2| − − 1 µ ρ 1φe 2φe 1φρ 2 ρ φe φe φρ ; − | 21| − − | 12| − 4 = ( sin ( 1 2 +21 ) 1 2 12 2 ( sin ( ) + sin ( + ))) =0 F.2. Supersymmetric vacuum manifold 195

µ Lν− ν τ τ e e φν φν φe φe Re | 2| 1 2 | 1|| 2| − − 1 µ ρ φν 1φν 2φρ 1 ρ 2 φν φν φρ ; − | 21| − − | 12| − (F.33) 4 = ( sin1( 2 21 + ) 1 2 12 Le− e ρ φe φe φρ ρ φe φe φρ ; − Im | 22| |( 21| sin ( − − )−|+ 12| sin ( − + ))) =0 Lν−1 ν ρ φν 1 φν 2 φρ 21 ρ φν 1 φν 2 φρ 12 ; − 2Im = | 2| (| 21| cos ( − − )−| 12| cos ( − + )) =0 (F.34) 1 1 2 21 1 2 12 and 2 = ( cos ( ) cos ( + )) =0 µ Le+ µ e ρ φρ τ2 e 2 ν 2 e − Re − | 1|| 11| 1 | 1| | 1| | 1| 1 µ e ρ φe φe11 φρ ρ φe φe φρ − | 2| | 21| − − | 12| − 16 = 16 cos ( ) +5 + τ τ e 2 ν 21 e 2 ν21 ν e φe1 φe2 φν12 φν ; 18 2 (| 2| cos| (2| | 1| | 1||) +2|| 2| cos ( − −+ )) 1 2 1 2 + 5 + +4 cos ( + ) =0 µ Lν+ µ ν ρ φρ τ2 ν 2 e 2 ν − Re − | 1|| 11| 1 | 1| | 1| | 1| 1 µ ν ρ φν φν11 φρ ρ φν φν φρ − | 2| | 21| − − | 12| − 16 = 16 cos ( ) +5 + τ τ ν 2 e 2 ν 1 e2 e 21ν φν φν1 φe2 φe12 ; 1 2 8| 2| ( | 2|cos|( 1| | 1|| 2|| )2+| cos−( − + )) (F.35) 1 2 1 2 + 5 + +4 cos ( + ) =0 Le+ e ρ φρ Im | 1|| 11| 1 e 11ρ φe φe φρ ρ φe φe φρ ; | 2| | 12| − −| 21| − − 2 = 2 sin ( ) 1 2 12 1 2 21 + ( sin ( + ) sin ( )) =0 Lν+ ν ρ φρ Im | 1|| 11| 1 ν ρ 11 φν φν φρ ρ φν φν φρ ; | 2| | 12| − −| 21| − − 2 = 2 sin ( ) (F.36) 1 2 12 1 e ;2 ν 21 where, as before,+ the( remainingsin ( eight+ real equations) sin for( V= 2 2 are)) obtained=0 by swapping ↔ . L− e Le− e Le− ν Lν− ν Lν− Focusing first on , one finds that | 1| | 2| and | 1| | 2| . 1 2 Le− Lν− Thus, we can consider just and as independent1 2 equations. For1 instance,2 from Le− + = 0 + = 0 Im one readily gets 1 1 1 ρ φe φe φρ =0 | 21| − : ρ φe1 φe2 φρ12 (F.37) | 12| − − cos ( 1 2 + 21 ) = LV− LV− cos ( ) Onφρ top ofφρ that, the remaining ρ equationsρ can be solved only for − , which, plugged into Eq. (F.37) gives | 12| ρ| 21|. Thus, we end up with the 12 21 Re = Im = 0 following condition for the off-diagonal entries of the matrix: = = ρ ρ∗ : 21 12 (F.38) Le− Lν− Inserting this into the Re and Re = equations, they simplify to 1 1 µ ρ τ τ ν ν ν e e ; − | 12| =0 1 2 | 1|| 2|=0 − (F.39) µ ρ τ τ e e ν e ν ; | 12| 1 2 | 1|| 2| − 4 = sin (Φ Φ ) cscΦ (F.40) 4 = sin (Φ Φ ) cscΦ Appendix F. Flipped SO vacuum 196 (10) where we have denoted ν φν φν φρ ; e φe φe φρ : ≡ − ≡ − (F.41) 1 2 12 1 2 12 These, taken together,Φ yield + Φ + e e e ν ν ν ; | 1|| 2| −| 1|| 2| (F.42) and sinΦ = sinΦ µ ρ ν e ν ν e e | 12| − : | 1|| 2| | 1|| 2| τ τ ν e (F.43) 1 2 − 4 sinΦ sinΦ + = ν e Notice− that in the zero phases limit the constraintsin (F.42)(Φ Φ is )trivially relaxed, while sin Φ ν sine Φ sin(Φ −Φ ) Ï . LV+ Returning1 to the equations, the constraint (F.38) implies, e.g. Le+ e ρ φρ ; =0Im | 1|| 11| Le+1 e ρ φρ11 ; | 2|| 22| Im = sin ( ) =0 Lν+2 ν ρ φρ22 ; | 1|| 11| Im = sin ( ) =0 Lν+1 ν ρ φρ11 : | 2|| 22| Im = sin ( ) =0 (F.44) 2 φρ 22 φρ For generic VEVs, these relationsρ = requireτ sin (and ) =0 to vanish. In conclusion, a 11 22 nontrivial vacuum requires (andD hence for consistency)SO to be hermitian. This is a consequencei i ∗ of the fact that -flatness for the flipped embedding implies h i , cf. Eq. (F.30). Let us also note that such a setting is preserved by (10) supersymmetric wavefunction renormalization. 16 = 16ρ ρ† LV+ ρ ν ν Taking in the remaining Re equations and trading | 12| for | 1|| 2| Le+ ; e e Lν+; in Re by means of Eq. (F.39) and for | 1|| 2| in Re via Eq. (F.40), one = = 0 obtains 1 2 1 2 = 0 = 0 µ Le+ e µρ τ2 ν 2 e 2 − Re | 1| − 11 1 | 1| | 1| 1 τ τ ν 2 e 2 τ τ ν ν e ν e ; 16 = 1 2 | 216| |+52| 1+2 | 1|| 2|| 2| + +5 +4 sinΦ cscΦ =0 µ Le+ e µρ τ2 ν 2 e 2 − Re | 2| − 22 2 | 2| | 2| 2 τ τ ν 2 e 2 τ τ ν ν e ν e ; 16 = 1 2 | 116| |+51| 1+2 | 1|| 2|| 1| + +5 +4 sinΦ cscΦ =0 µ Lν+ ν µρ τ2 e 2 ν 2 − Re | 1| − 11 1 | 1| | 1| 1 τ τ e 2 ν 2 τ τ ν e e ν e ; 16 = 1 2 | 216| |+52| 1+2 | 2|| 1|| 2| + +5 +4 cscΦ sinΦ =0 µ Lν+ ν µρ τ2 e 2 ν 2 − Re | 2| − 22 2 | 2| | 2| 2 τ τ e 2 ν 2 τ τ ν e e ν e : 16 = 1 2 | 116| |+51| 1+2 | 1|| 1|| 2| (F.45) + +5 +4 cscΦ sinΦ =0 F.2. Supersymmetric vacuum manifold 197

Since only two out of these four are independent constraints, it is convenient to consider the following linear combinations

C ν 2 e Le+ e Le+ e 2 ν Lν+ ν Lν+ ; 3 ≡| 1| | 1|Re −| 2|Re −| 1| | 1|Re −| 2|Re (F.46) 1 2 1 2 C ν 2 e Le+ e Le+ e 2 ν Lν+ ν Lν+ ; 4 ≡| 2| | 1|Re −| 2|Re −| 2| | 1|Re −| 2|Re (F.47) 1 2 1 2 which admit for a simple factorized form

µC ν 2 e 2 ν 2 e 2 3 | 2| | 1| −| 1| | 2| τ2 ν 2 e 2 τ τ ν 2 e 2 µρ ; 16 ×= 2 | 2| | 2| 1 2 | 1| | 1| − 22 (F.48) µC ν 2 e 2 ν 2 e 2 4 5| 2| | 1| +−| 1| +| 2| + 16 =0 τ2 ν 2 e 2 τ τ ν 2 e 2 µρ : 16 ×= 1 | 1| | 1| 1 2 | 2| | 2| − 11 (F.49) These relations can be5 generically+ satisﬁed+ only if the+ square16 brackets=0 are zero, providing

µρ τ2 ν 2 e 2 τ τ ν 2 e 2 ; 11 1 | 1| | 1| 1 2 | 2| | 2| µρ τ2 ν 2 e 2 τ τ ν 2 e 2 : 16 22 =5 2 | 2| + | 2| + 1 2 | 1| + | 1| (F.50) ~r ν ; e ~r 16 =5 + + + Byν introducing; e a pair of symbolic 2-dimensional vectors 1 | 1| | 1| and 2 | 2| | 2| one can write = ( ) = r2 ν 2 e 2 ; ( ) 1 | 1| | 1| r2 ν 2 e 2 ; | 2| | 2| ~r2:~r= ν +ν e e : 1 2 | 1|| 2| | 1|| 2| = + (F.51) which, in combination with eqs. (F.43)= and (F.50)+ yields µ ρ τ ρ τ r2 22 1 − 11 2 ; 1 − τ2τ 2µ(ρ τ 1 25ρ τ ) r2 = 11 2 − 22 1 ; 2 − 3τ τ2 1 2 2 µ( ρ 5 ν ) e ~r :~r= | 12| − : 1 2 τ τ 3 ν e (F.52) 1 2 − 4 sinΦ sinΦ = sin Φ Φ With this at hand,α theα vacuum manifold can be( conveniently) parametrized by means of two angles 1 and 2 ν r α ; e r α ; | 1| 1 1 | 1| 1 1 ν r α ; e r α : | 2| 2 2 | 2| 2 2 = sin = cos (F.53) = sin = cos Appendix F. Flipped SO vacuum 198

α(10)± α α which are fixed in terms of the superpotential parameters. By defining ≡ 1 ± 2, Eqs. (F.51)–(F.53) give ~r :~r ν e α− 1 2 ξ − ; r r ν e (F.54) 1 2 − sinΦ sinΦ cos = = where ρ sin (Φ Φ ) ξ | 12| : ρ τ ρ τ 5 5 ρ ρ (F.55) τ2 τ2 − 11 2 −6 22 1 22 11 = q 1 2 Analogously, Eq. (F.42) can be rewritten as + 26 α α e α α ν ; 1 2 − 1 2 (F.56) cos cos sinΦ = sin sin sinΦ which gives e α α− + − ; ν α− α+ (F.57) sinΦ cos cos = and thus, using Eq. (F.54), we obtainsinΦ cos + cos ν e α+ ξ : ν e − (F.58) sinΦ + sinΦ cos =ν e α+ α− ξ Notice also that in the real case (i.e., sin (Φ) Φis) undetermined, while . This justifies the shape of the vacuum manifold given in Eq. (4.21) of Sect. 4.4.2. Φ =Φ =0 cos = F.3 Gauge boson spectrum

In order to determine the residual symmetry corresponding to a specific vacuum con- SO U X figuration we compute explicitly the gauge spectrum. Given the ⊗ covariant derivatives for the scalar components of the Higgs chiral superfields (10) (1) Dµ ∂µ ig Aµ ij +ij igXXµ ; − ( ) ( ) − Dµ ∂µ ig Aµ ij −ij igXXµ ; 16 = 16 − ( )( )Σ( )16 16 Dµ ∂µ ig Aµ ij +ij ; ; 16 = 16 − ( )( )Σ (16) + 16 (F.59) h i 45 =ij 45 ( ) Σ 45 where the indices in brackets stand for ordered pairs, and the properly normalized kinetic terms † Dµ †Dµ ;( ) Dµ Dµ ; 1 Dµ †Dµ ; 4 Tr (F.60) one can write the 46-dimensional16 16 gauge16 boson16 mass matrix45 governing45 the mass bilinear of the form µ µ T Aµ ij ;Xµ 2 A; X A kl ;X ( ) M ( ) (F.61) 1 ( ) ( ) ( ) 2 F.3. Gauge boson spectrum 199 as

2ij kl 2ij X 2 A; X M( )( ) M( ) : M X2 kl XX2 ! (F.62) M ( ) M ( ) = The relevant matrix elements are given by

† 2ij kl g 2 † +ij ; +kl −ij ; −kl M( )( ) h i { ( ) ( )}h i { ( ) ( )} ; † ; ; = 16 Σ Σ 16 + 16 Σ Σ 16 +ij h i +kl h i Tr ( ) ( ) 1 h i h i + Σ 45 Σ 45 2

† 2ij X ggX † +ij −ij ; M( ) h i ( ) h i − ( ) † X2 kl ggX † +kl −kl ; M ( ) =2 h16 i Σ ( ) h16 i − 16 Σ ( ) 16 † XX2 gX2 † : M =2 h i16h Σi 16 16 Σ 16 (F.63) =2 16 16 + 16 16 F.3.1 Spinorial contribution

Considering ﬁrst the contribution of the reducible representation 1 ⊕ 2 ⊕ 1 ⊕ 2 to the gauge boson mass matrix, we ﬁnd 16 16 16 16 2 ; ; ; M16 1 (F.64) 2 ; ; 45 ; M16(1 3 0)15 =0 (F.65) 45 (8 1 0) =0

2 ; ; 1 M16 − 3 15 45 g 2 e 2 ν 2 e 2 ν 2 e 2 ν 2 e 2 ν 2 ; (3 1 ) = | 1| | 1| | 2| | 2| | 1| | 1| | 2| | 2| (F.66) − ; + + + + + + + + In the 45 45 basis (see Table 4.5 for the labelling of the states) we obtain

2 ; (6; 1 6 ) M16 6 g 2 ν 2 ν 2 ν 2 ν 2 ig 2 e∗ν e∗ν ν∗e ν∗e (3 2 + ) =| 1| | 2| | 1| | 2| − 1 1 2 2 1 1 2 2 ; ig 2 e ν∗ e ν∗ ν e∗ ν e∗ g 2 e 2 e 2 e 2 e 2 ! (F.67) 1 2 1 2 | 1| | 2| | 1| | 2| 1 + 2 + 1 + 2 ( + + + ) ( + + + ) + + + Appendix F. Flipped SO vacuum 200 (10) ; − ; 0 ; + ; The ﬁve dimensional SM singlet mass matrix in the 45 45 45 45 1 basis reads 15 1 1 1 1 2 ; ; M16

3 g 2S i√ g 2S 3 g 2S i√ g 2S∗ √ ggXS (1 1 0) = 2 1 3 − 2 2 − 3 − 1  i√ g 2S∗ g 2S q iggXS  − 3 3 1 3 3 3  3 g 2S g 2S √ ggXS   − 2 1 2  (F.68)  √ 32 0 0 2   i q g 2S g 2S iggXS∗   3 0 0 1 − 2 3   √ ggXS iggXS∗ √ ggXS iggXS gX2 S   − 1 − 2 3 1   3 03 0 2 

S e 2 e3 2 ν 2 2 ν 2 e 2 e 2 2ν 2 ν 2 S2 e 2 e 2 where 1 ≡ | 1| | 2| | 1| | 2| | 1| | 2| | 1| | 2| , 2 ≡ | 1| | 2| − ν 2 ν 2 e 2 e 2 ν 2 ν 2 S e ν∗ e ν∗ e∗ν e∗ν | 1| −| 2| | 1| | 2| −| 1| −| 2| and 3 ≡ 1 1 2 2 1 1 2 2. + + + ; ; + + + + + M2 For generic+ VEVs+ Rank 16 , and we recover+ 12+ massless+ gauge bosons with the quantum numbers of the SM algebra. (1 1 0)=4 We veriﬁed that this result is maintained when implementing the constraints of the ﬂipped vacuum manifold in Eq. (4.21). Since it is, by construction, the smallest algebra that can be preserved by the whole vacuum manifold, it must be maintained when H h i adding the contribution. We can therefore claim thatH the invariant algebra on the generic vacuum is the SM. On the other hand, the plays already an active 45 role in this result since it allows for a misalignment of the VEV directions in the two H H 45 SU U ⊕ spinors such that the spinor vacuum preserves SM and not ⊗ . More details shall be given in the next section. 16 16 (5) (1)

F.3.2 Adjoint contribution

H Considering the contribution of h i to the gauge spectrum, we ﬁnd

452 ; ; ; M45 1 (F.69) 2 ; ; 45 ; M 15 45(1 3 0) =0 (F.70) 2 ; ; 1 45 g 2ωB2 L : M45 − 3 15 − (F.71) (8 1 0) 45=0 − ; + (3 1 ) =4 Analogously, in the 45 45 basis, we have

2 ; ; 1 (6 6 ) M45 6 g 2 ωR ωB L 2 ω−ω+ i √ g 2ωB Lω− − − : (3 2 + ) = √ i g 2ωB Lω+ g 2 ωR ωB L 2 ω−ω+ ! (F.72) − − − − ( + ) +2 2 2 2 2 ( ) +2 F.3. Gauge boson spectrum 201

; − ; 0 ; + ; The SM singlet mass matrix in the 45 45 45 45 1 basis reads 15 1 1 1 1 g ω ω−ω i g ωRω− g ω−  2 R2 + − 2 2 2  ; ; 00i g ωRω g ω 0−ω i g 00ωRω− : M2  2 + 2 + − 2  45  0 4 + 4 4 ( ) 0  (F.73)  g 2 ω+ 2 i g 2ωRω+ g 2 ωR2 ω−ω+    (1 1 0) =  0 4 8 4 0     0 4 ( ) 4 4 + 0  002 ; ; 0 00 For generic VEVs we ﬁnd Rank M45 leading globally to the 14 massless SU C SU L U 3 gauge bosons of the ⊗ ⊗ algebra. (1 1 0) = 2 F.3.3 Vacuum little(3) group (2) (1)

With the results of sections F.3.1 and F.3.2 at hand the residual gauge symmetry can be readily identiﬁed from the properties of the complete gauge boson mass matrix. For the sake of simplicity here we shall present the results in the real VEV approximation. Trading the VEVs for the superpotential parameters, one can immediately identify the strong and weak gauge bosons of the SM that, as expected, remain massless:

2 ; ; ; M 15 2 ; ; 45 : M 1 (F.74) (8 1 0) 45 =0 Similarly, it is straightforward to obtain(1 3 0) =0

2 ; ; 1 M − 15 3 g 45 2 µ ρ τ τ τ ρ τ τ τ ρ τ ρ τ : (3 1 ) = 2 τ2τ2 22 1 1 − 2 11 2 2 − 1 22 1 11 2 (F.75) 41 2 3 ( (5 ) + (5; ; )) +2 ( ; +; ) 9 2 1 2 On the other hand, the complete matrices M 6 and M turn out to be quite involved once the vacuum constraints are imposed, and we do not show them (3 2 + ) (1 1 0) here explicitly. Nevertheless, it is suﬃcient to consider

g 2 2 ; ; 1 µ2 r2 r2 τ2r4 τ2r4 τ τ r2r2 α− Tr M 6 µ2 1 2 1 1 2 2 1 2 1 2 (F.76) (3 2 + ) = 16 + + + + (1+cos2 ) and 8

g 4r2r2 2 ; ; 1 1 2 µ4 µ2 τ2r2 τ2r2 det M 6 µ4 1 1 2 2 τ τ r r α− α− (3 2 + ) = 512 + 32 + 2 2 2 2 − 2 128 1 2 1 2 (F.77) + (1 cos2 ) sin Appendix F. Flipped SO vacuum 202 (10) α− 2 ; ; 1 M 6 to see that for a generic non-zeroα− value of one getsr Rank . On the other hand, when (i.e., h 1i ∝h 2i) or 2 (i.e., h 2i ), 2 ; ; 1 sin ; ; (31 2 + ;)=2; 1 Rank M 6 and one is left with an additional massless 6 ⊕ − 6 = 0 16 16 = 0 16 = 0 gauge boson, corresponding to an enhanced residual; ; symmetry. (3 2 + )=1 M2 (3 2 + ) (3 2 ) In the case of the α-dimensional− matrix it is sufficient to notice that for a generic non-zero 5 2 ; ; (1 1 0); Rank M (F.78) sin U Y on the vacuum manifold, which leaves a massless(1 1 0)=4 gauge boson, thus completing α− r 2 ; ; the SM algebra. As before, for or for 2 , we find Rank M . ; ; (1)1 ; ; 1 Taking into account the massless states in the 6 ⊕ − 6 sector, we recover, SU U = 0 = 0 (1 1 0) = 3 as expected, the flipped ⊗ algebra. (3 2 + ) (3 2 ) (5) (1) Appendix G

E vacuum

G.16 The SU formalism

Following closely the notationE of Refs. [201,SU 224],C SU we decompoL SU seR the adjoint and fundamental representations(3) of 6 under its ⊗ ⊗ maximal subalgebra as ; ; ; ; (3); ; (3); ; (3); ; ≡ ⊕ ⊕ ⊕ ⊕ α i i′ α ij′ Tβ Tj Tj Qij Qα ; ⊂ ⊕ ⊕ ′ ⊕ ′ ⊕ (G.1) 78 (8 1 1) (1 8 1) (1 1 8) (3 3 3)i (3 αj3′ 3) ; ; ; ; ; ; vαi vj v ; ≡ ⊕ ⊕ ≡ ⊕ ′ ⊕ (G.2) αi j′ ; ; ; ; ; ; u ui uαj′ ; 27 ≡ (3 3 1) ⊕ (1 3 3) ⊕ (3 1 3) ≡ ⊕ ⊕ (G.3) SU C SU L where the greek,27 latin and(3 3 primed-latin1) (1 3 3) indices,(3 1 3) corresponding to , and SU R SU , respectively, run from to . As far as the algebras in Eq. (G.1) are (3) (3) concerned, the generators follow the standard Gell-Mann convention (3) 1 3 i (3) T(1) 1 T1 T2 ; T(2) T1 T2 ; 2 2 1 2 2 − 1 T(3) 1 T1 T2 ; T(4) 1 T1 T3 ; 1 − 2 3 1 = 2i ( + ) = 2 ( ) (G.4) T(5) T1 T3 ; T(6) 1 T2 T3 ; 3 − 1 3 2 = 2i ( ) = 2 ( + ) T(7) T2 T3 ; T(8) √1 T1 T2 T3 ; = 2 ( 3 − 2 ) = 2 ( 3 +1 ) 2 − 3 Ta k δkδa T a T b δab b l b l = ( ) = ( + ( ) (2) ) 1 with , so they are all normalizedE so that Tr 2 . Taking into account Eqs. (G.1)–(G.4), the 6 algebra can be written as ( ) = Tα;Tγ δαTγ δγ Tα = β η η β − β η i k i k k i Tj ;Tl δl Tj δj Tl [ ] = − Ti′ ;Tk′ δi′ Tk′ δk′ Ti′ j′ l′ l′ j′ j′ l′ [ ] = − Tα;Ti Tα;Ti′ Ti;Ti′ ; β j β j′ j j′ [ ] = (G.5) [ ] = [ ] = [ ]=0 Appendix G. E vacuum 204 6

Qγ ;Tα δγ Qα ij′ β β ij′

ij′ α α ij′ Qγ ;Tβ δγ Qβ [ ] = − Qγ ;Tk δkQγ ij′ l − i lj′ [ ij′ k] = i kj′ Qγ ;Tl δl Qγ [ γ ] = γ Q ;Tk′ δk′ Q ij′ l′ j′ il′ [ ] = − ij′ k′ j′ ik′ Qγ ;Tl δl Qγ ; [ ′ ] = ′ (G.6) [ ] = Qα ; Qkl′ δαδkTl′ δαδl′ Tk δkδl′ Tα ij′ β − β i j′ − β j′ i i j′ β α β αβγ pq′ Q ; Q ikpj l q Q ij′ kl′ ′ ′ ′ γ [ ij′ kl′] = ikp j′l′q′ γ + Q ; Q αβγ Q ; α β pq′ [ ] = − (G.7) [ ] = The action of the algebra on the fundamental representation reads β β Tγ vαi δαvγi27 k k Tl vαi δi vαl k′ = T vαi l′ β = β k Q vαi δ pikv pq′ α q′

pq′ =0 p γq′ Qβ vαi δi βαγv ; = (G.8) = Tβvi γ j′ Tkvi δivk l j′ l j′ =0− Tk′ vi δk′ vi l′ j′ j′ l′ β =i i βk′ Q v δ q j k v pq′ j′ − p ′ ′ ′ pq′ i= q′ pik Qβ vj δj vβk ; ′ = ′ (G.9) = Tβvαj′ δαvβj′ γ − γ k αj′ Tl v = j Tk′ vαj′ δ ′ vαk′ l′ − l′ β αj=0′ j′ βαγ Q v δ vγp pq′ q′ = − pq′ αj′ α q′j′k′ p Qβ v δβ vk ; = − ′ (G.10) = G.1. The SU 3 formalism 205 (3) and accordingly on Tβuαi δαuβi 27 γ − γ Tkuαi δiuαk l − l Tk′ uαi = l′ β αi= i βαγ Q u δ uγq pq′ − p ′ pq′ αi=0 α pik q′ Qβ u δβ uk ; = − (G.11) = β j′ Tγ ui

k j′ k j′ Tl ui δi ul j =0 j Tk′ u ′ δ ′ uk′ l′ i − l′ i β j=′ j′ βk Q u δ piku pq′ i − q′ pq′ j=′ p q′j′k′ Qβ ui δi uβk′ ; = (G.12) = β β Tγ uαj′ δαuγj′ k Tl uαj′ Tk′ u = δk′ u l′ αj′ j′ αl′ β =0 β k′ Q uαj δ q j k u pq′ ′ α ′ ′ ′ p pq′ = q′ γp Qβ uαj′ δj βαγ u : = ′ (G.13) = Given the SM hypercharge deﬁnition

Y TL(8) TR(3) TR(8) ; √ √ (G.14) 1 1 = + + the SM-preserving vacuum direction3 corresponds to3 [201] a a b ′ ′ ′ ′ ′ ′ ′ a T a T 3 T T T 4 T T 3 T T T ; 3′ 2′ 1′ 2′ 3′ 1′ 2′ 1 2 3 h i 1 2 2 3 √ 1 2 − 3 √ 1 − 2 √ 1 2 − 3 (G.15) 78 = + + ( + 2 ) + ( ) + ( + 2 ) ev νv ; eu ′ νu ′ ; 6 3′ 3′ 2 3 2 6 h i 3 2 3 3 (G.16) a a a a b e e ν ν where 1, 2, 3, 4, 3,27, =, and+ are SM-singlet27 = VEVs.+a ; a This canb be checked by 3 4 3 meansa a∗ of Eqs. (G.5)–(G.13). Notice that the adjoint VEVs and are real, while 1 2. The VEVs of ⊕ are generally complex. = 27 27 Appendix G. E vacuum 206 6 G.2 E vacuum manifold

D Working out6 the -ﬂatness equations, one ﬁnds that the nontrivial constraints are given by a a a∗ a∗ DEα 3 4 a∗ a 3 4 e∗ν e ν∗ e∗ν e ν∗ ; √ − √ − 1 √ − √ 1 − 1 2 − 2 2 1 1 2 2 3 3 DT = a 2 a 2 e 2 e 2 + e 2 e 2 + ν 2 ν 2=0ν 2 ν 2 ; R |61| −|2 2| | 1| −|6 1| 2 | 2| −| 2| | 1| −| 1| | 2| −| 2| (8) DT a 2 a 2 ν 2 ν 2 ν 2 ν 2 ; R =3| 2| −| 1| | +21| −| 1| | 2| +2−| 2| + + =0 (3) DT e 2 ν 2 e 2 ν 2 e 2 ν 2 e 2 ν 2 ; L = | 1| | 1| +| 2| | 2| −|+ 1| −| 1| −|=0 2| −| 2| (G.17) (8)

DEα ; ; where= is+ the ladder+ operator+ from the sub-multiplet=0 of . Notice that the DT DT DT relations corresponding to R , R and L are linearly dependent, since the linear (8) (3) (1(8) 1 8) 78 combination associatedW toH the SM hypercharge in Eq. (G.14) vanishes. The superpotential in Eq. (4.41) evaluated on theF vacuum manifold (G.15)-(G.16) yields Eq. (4.52). Accordingly, one finds the following -flatness equations Fa µa τ e ν τ e ν ; 2 − 1 1 1 − 2 2 2 Fa1 µa τ ν e τ ν e ; 1 − 1 1 1 − 2 2 2 2 = =0 Fa µa τ ν ν e e τ ν ν e e ; = 3 − √ 1 1 1 − =01 1 2 2 2 − 2 2 3 1 Fa = µa (τ (ν ν τ2ν ν ) + ;( 2 )) =0 4 √6 1 1 1 2 2 2 4 1 F = µb + ( τ ν ν + e e ) =0τ ν ν e e ; b 2 3 − r 1 1 1 1 1 2 2 2 2 2 F 3 ρ e 2ρ e τ √ b a e a ν ; e = ( ( + ) + ( + )) =0 11 1 3 12 2 − 1 3 − 3 1 1 1 1 Fe ρ e ρ e τ √ b a e a ν ; 3 = 3( 21 1 + 22 2) − 2 6 ( 3 − 3) 2 +3 1 2 =0 2 Fν ρ ν ρ ν τ √ √ a a √ b ν a e ; 3 = 3( 11 1 + 12 2 ) − 1 6 ( 3 −) 4+3 3 =01 2 1 1 Fν ρ ν ρ ν τ √ √ a a √ b ν a e ; 6 = 6( 21 1 + 22 2) − 2 2( 3 3 − 3 4 +2 3 3) 2 +6 2 2 =0 2 Fe ρ e ρ e τ √ b a e a ν ; 6 = 6( 11 1 + 21 2) − 1 2( 3 − 3 3 1 +2 2 31 ) +6 =0 1 Fe ρ e ρ e τ √ b a e a ν ; 3 = 3( 12 1 + 22 2) − 2 6 ( 3 − 3) 2 +3 2 2 =0 2 Fν ρ ν ρ ν τ √ √ a a √ b ν a e ; 3 = 3( 11 1 + 21 2 ) − 1 6 ( 3 −) 4+3 3 =01 1 1 1 Fν ρ ν ρ ν τ √ √ a a √ b ν a e : 6 = 6( 12 1 + 22 2) − 2 2( 3 3 − 3 4 +2 3 3) 2 +6 1 2 =0 (G.18) 2 6 = 6( + ) 2( 3 3 +2 3 ) +6 =0 G.2. E vacuum manifold 6 207 aFollowinga a a the strategyb of Appendix F.2 one can solve the first five equations above for 1, 2, 3, 4 and 3: µa τ ν e τ ν e ; 1 1 1 1 2 2 2 µa τ e ν τ e ν ; 2 1 1 1 2 2 2 √ µa = τ ν ν + e e τ ν ν e e ; 3 = 1 1 1+− 1 1 2 2 2 − 2 2 √ µa τ ν ν τ ν ν ; 6 4 = − (1 1 1 −2 2 2 )2+ ( 2 ) √ µb √ τ ν ν e e τ ν ν e e : 2 3 = 1 1 1 1 1 2 2 2 2 2 (G.19) a a τ τ ∗ 3 = 2 + + + Since 1 2 and 1 and 2 can be( taken( real without) ( loss of generality)) (see Sect. 4.5.2), the first two equations above imply = τ ν e τ ν e τ e ν ∗ τ e ν ∗ ; 1 1 1 2 2 2 1 1 1 2 2 2 (G.20) F Using Eq. (G.19) the remaining+ -flatness= conditions( ) + in( Eq.) (G.18) can be rewritten in the form a µFe µ ρ e ρ e τ2 ν ν e e e 11 1 12 2 − 1 1 1 1 1 1 1 τ τ ν ν e ν ν e e e ; − 1 2 2 1 2 2 2 2 2 1 3 =3 ( + ) 4 ( + )

a (3 + ( +4 ) ) =0 µFe µ ρ e ρ e τ2 ν ν e e e 11 1 21 2 − 1 1 1 1 1 1 1 τ τ ν ν e ν ν e e e ; − 1 2 2 1 2 2 2 2 2 1 3 =3 ( + ) 4 ( + )

a (3 + ( +4 ) ) =0 µFν µ ρ ν ρ ν τ2 e e ν ν ν 11 1 12 2 − 1 1 1 1 1 1 1 τ τ e e ν e e ν ν ν ; − 1 2 2 1 2 2 2 2 2 1 3 =3 ( + ) 4 ( + )

a (3 + ( +4 ) ) =0 µFν µ ρ ν ρ ν τ2 e e ν ν ν 11 1 21 2 − 1 1 1 1 1 1 1 τ τ e e ν e e ν ν ν ; − 1 2 2 1 2 2 2 2 2 1 3 =3 ( + ) 4 ( + ) (G.21) (3 + ( +4 ) ) =0 and the additional four relations canD be again obtained by exchanging ↔ . Similarly, the triplet of linearly independent -ﬂatness conditions in Eq. (G.17) can be brought to 1 2 the form

DEα e∗ν e ν∗ e∗ν e ν∗ ; 1 1 − 1 1 2 2 − 2 2 DT ν 2 ν 2 ν 2 ν 2 ; R | 1| −| 1| | 2| −| 2| (3) = + =0 DT e 2 ν 2 e 2 ν 2 e 2 ν 2 e 2 ν 2 : L = | 1| | 1| + | 2| | 2| −|=0 1| −| 1| −| 2| −| 2| (G.22) (8) = + + + =0 Appendix G. E vacuum 208 6

D e∗; e ; Combining these with Eq. (G.20), the -ﬂatness is ensured if and only if 1 2 1 2 and ν∗; ν ; SO 1 2 1 2. Hence, in complete analogy with the ﬂipped case Eq. (F.30), one can = write = (10)

iφe iφe e ; e ; e ; ; e ; e ; e− ; ; 1 2 ≡| 1 2| 1 2 ≡| 1 2| iφν 1 2 iφν 1 2 ν ; ν ; e ; ; ν ; ν ; e− ; : 1 2 ≡| 1 2| 1 2 ≡| 1 2| 1 2 1 2 (G.23)

From now on,SO the discussion of the vacuum manifold follows very closely that for the ﬂipped in Sect. F.2 and we shall not repeat it here.ρ τ In particular the existence of a nontrivial vacuum requires the hermiticity of the and couplings. This (10) D F i i ∗ is related to the fact that - and -ﬂatness require h i . The detailed shape of the resulting vacuum manifold so obtained is given in Eq. (4.53) of Sect. 4.5.2. 27 = 27

G.3 Vacuum little group

In order to find the algebra left invariantE by the vacuum configurations in Eq. (4.53), we need to compute the action of the 6 generators on the ⊕ 1 ⊕ 2 ⊕ 1 ⊕ 2 VEV. From Eqs. (G.5)–(G.6) one obtains 78 27 27 27 27 Tα β h i i b i i i i i i Tj 3 δ Tj1 δj1T δ Tj2 δj2T δ Tj3 δj3T h78i =0√ 1 − 1 2 − 2 − 3 3 a i′ i′ ′ ′ i′ i′ ′ ′ i′ i′ ′ ′ i′ i′ ′ ′ i′ T a δ T δ T a δ T δ T 4 δ T δ T δ T δ T j′ 78 = ( ′ j3′ j3′ ′ + ′ j2′ j2′ 2 ′ +2 ′ )j1′ j1′ ′ ′ j2′ j2′ ′ h i 16 2 − 2 2 3 − 3 √ 1 − 1 − 2 2 a i′ ′ ′ i′ i′ ′ ′ i′ i′ ′ ′ i′ 3 δ T δ T δ T δ T δ T δ T 78 = ( ′ j1′ j1′ )′ + (′ j2′ j2′ ′ ) + ′ j3′ ( j3′ ′ + ) √ 1 − 1 2 − 2 − 3 2 3 a α ′ α ′ α ′ α ′ α ′ α Q a δ Q a δ Q 3 δ Q δ Q δ Q ij′ + ( j3′ i ′ +j2′ i ′ j1′ i 2′ j2′ +2i ′ j3′) i ′ h i −61 2 − 2 3 − √ 1 2 − 3 a b ′ α ′ α α α α 4 δ Q δ Q 3 δ Q δ Q δ Q 78 = ( j1′ i ′ ) j2′ (i ′ ) i1 ( j′ i2 + j′ i3 2j′ ) − √ 1 − 2 − √ 6 1 2 − 3 a ij′ j′ i ′ j′ i ′ j′ i ′ j′ i ′ j′ i ′ Q a δ Q a δ Q 3 δ Q δ Q δ Q α ( ′ α3 ′ )α2 ( ′ +α1 ′ α22 ′) α3 h i 12 2 2 3 6√ 1 2 − 3 a b j′ i ′ j′ i ′ i j′ i j′ i j′ 4 δ Q δ Q 3 δ Q δ Q δ Q ; 78 = ( ′ α1) + (′ α2 ) + ( α1 + α2 2 α3 ) √ 1 − 2 √ 61 2 − 3 (G.24) + ( ) + ( + 2 ) 2 6 G.3. Vacuum little group 209 on the adjoint vacuum. For h 1 ⊕ 2i one finds α Tβ h 1 ⊕ 2i 27 27 Ti e e δ vi ν ν δ vi j j3 ′ j3 ′ h271 ⊕ 272i =0− 1 2 3 − 1 2 2 Ti′ e e δi′ v ν ν δi′ v j′ ′ j3′ ′ j3′ h271 ⊕ 272i = (1 +2 )[3 ] (1 +2 )[2 ] Qα e e δ vαk′ ν ν δ vαk′ ij′ i3 j′ ′k′ i3 j′ ′k′ h271 ⊕ 272i =− ( +1 )[2 ] +3 ( +− )[ 1 ]2 2 ij′ j′ i k j′ i k Q e e δ vαk ν ν δ vαk ; α ′ 3 ′ 3 h271 ⊕ 272i = (1 +2 )[3 ] 1 ( +2 2)[ ] (G.25) 27 27 = ( + )[ ] + ( + )[ ] and, accordingly, for 1 ⊕ 2 α Tβ 27 27 1 ⊕ 2 i i ′ i ′ Tj e e δ uj3 ν ν δ uj2 271 ⊕ 272 =0 1 2 3 1 2 3 Ti′ e e δ ′ ui′ ν ν δ ′ ui′ j′ j3′ j2′ 1 ⊕ 2 − 1 2 3 − 1 2 3 α 27 27 = ( + )[ ′ ] + ( αk + )[ ] ′ αk Q e e δ i ku ν ν δ i ku ij′ j3′ j2′ 271 ⊕ 272 = −( 1 + 2)[ 3] ( −+ 1 )[ 2 ] 3 Qij′ e e δi j′ ′k′ u ν ν δi j′ ′k′ u : α 3 αk′ 2 αk′ 271 ⊕ 272 = (1 +2 )[3 ] ( 1 + 2 )[ 3 ] (G.26)

On the vacuum27 manifold27 = ( in+ Eq.)[ (4.53) one] ﬁnds + ( that+ )[ the generat]ors generally preserved by the VEVs of ⊕ 1 ⊕ 2 ⊕ 1 ⊕ 2 are

TC(1) 78TC(2) T27C(3) TC(4)27TC(5) T27C(6) TC(7)27TC(8) ; ; ;

TL(1) TL(2) TL(3) ; ; ; Y ; ; ; : (8 1 0) Qα Qα Q :′ Q (1 3′ 0) ; ; ; ; ; ′ ′ α11 α21 5 5 11: (1 211 0) 6 ⊕ − 6 (G.27) SU which generate an algebra. As an: (example3 2 + ) showing(3 2 the) nontrivial constraints enforced by the vacuum manifoldQα in Eq. (4.53), let us inspect the action of one of the (5) ′ lepto-quark generators, say 11 : Qα a √ a b Qα ; ′ √1 ′ 11 h i − 6 3 4 3 11 (G.28) Qα ; ′ 11 h781 ⊕= 2i + 3 + Qα : ′ 11 271 ⊕ 272 =0

a √27a 27b =0 It is easy to checkα that 3 4 3 vanishes on the whole vacuum manifold in Eq. Q U Y ′ (4.53) and, thus, 11 is preserved. Let us also remark that the charges above +SO3 + correspond to the standard embedding (see the discussion in sect. 4.5.2). In the SO ; ; (1) ﬂipped embedding, the ⊕ generators in Eq. (G.27) carry hypercharges 1 (10) ∓ 6 , respectively. (10) (3 2) (3 2) Appendix G. E vacuum 210 6

SOConsidering instead the vacuum manifold invariant with respect to the flipped hyperchargeQ (seeα Q Eqs.α Q (4.46)–(4.48)),′ Q ′ the preserved generators, in addition to ′ ′ α13 α23 those of the SM, are 13 23 ; ; . These,; ; for the standard hypercharge embed- (10) − 1 ⊕ 1 ding of Eq. (4.37), transform as 6 6 , whereas; ; with the; ; flipped hyper- 5 ⊕ − 5 charge assignment in Eq.SU (4.38),( the3 2 same) transform(3 2 + ) as 6 6 . Needless to say, one finds again as the vacuum little group. α (α3 2 + ) (3 2 ) It is interesting to consider the configuration 1 2 , which can be chosen (5) without loss of generality once a pair, let us say 2 ⊕ 2, is decoupled or when the H H = = 0 two copies of ⊕ are aligned. According to Eq. (4.53) this implies all VEVs equal a b e e 27 27 to zero but 3 − 3 and 1 ( 2). Then, from Eqs. (G.24)–(G.26), one verifies that the 27 27 preserved generators are (see Eq. (G.4) for notation) = TC(1) TC(2) TC(3) TC(4) TC(5) TC(6) TC(7) TC(8) ; ; ;

TL(1) TL(2) TL(3) ; ; ; : (8 1 0) TR(1) TR(2) TR(3) ; ; ; ; ; ; ; : (1 3 −0) ⊕ ⊕ TL(8) TR(8) ; ; ; : (1 1 1) (1 1 0) (1 1 +1) (G.29) + : (1 1 0) Qα Qα Q ′ Q ′ ; ; ; ; ; ′ ′ α11 α21 5 5 11 21 6 ⊕ − 6 Qα Qα Q ′ Q ′ ; ; ; ; ; ′ ′ α12 α22 1 1 12 22 : (3 2 +− 6) ⊕ (3 2 6) Qα Q ′ ; ; ; ; ; ′ α33 2 2 33 − 3:⊕ (3 2 ) 3 (3 2 + ) (G.30) SO a b SO U : (3 1 ) (3 1 + ) which support anU algebra. In particular, 3 − 3SOpreserves ⊗ , where the extra generator, which commutes with all generators, is pro- TL(8) TR(8)(10) =e TL(8) TR(8) (10) (1) − 1 − portional to (1) . On the other hand, the VEV breaks(10) E (while preserving the sum). We therefore recover the result of Ref. [201] for the 6 setting with H H H ⊕ ⊕ . 78 27 27 Bibliography

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I warmly thank: Stefano Bertolini for his attitude (would you ever bet there was something to dig out ?), Michal Malinský for teaching me a lot, Goran Senjanović for in- spiration, Ketan Patel for useful communications, Lorenzo Seri still looking forward for his ﬁrst symphony, Enzo Vitagliano for showing us it was possible to go barefoot all the summer, Goﬀredo Chirco as an example of how big things a small man can do, Adriano Contillo for his communicative skills, Marco Nardecchia because he is an happy person, Giorgio Arcadi for being always sober, Robert Ziegler for relaxed discussions about physics, Chetan Krishnan because he is not cynic at all, Jarah Evslin for a discussion on spinors and a proof of the roundedness of the earth, Chica and Smilla two most beautiful creatures, Paolo Antonelli for sharing some good time here in Trieste, Antonella Garzilli for being a bit lunatic, Angus Prain for initiating me to the spirit of the game, Daniele De Martino for telling us the importance of being present, Fabio Caccioli because it was nice to look at him listening to people, Taskin Deniz for the power of obsession, Maya Sundukova, Pierpaolo Vivo and Luca Tubiana for standing me and my hairs, Ugo Marzolino for being around me in the last 15 years and DFW for the footnotes.