Minimal SU(5) Asymptotic Grand Unification
Giacomo Cacciapaglia, Alan S. Cornell, Corentin Cot, Aldo Deandrea
PPrepared for submission to JHEP
Minimal SU(5) Asymptotic Grand Unification
Giacomo Cacciapaglia Alan S. Cornell Corentin Cot Aldo Deandrea Universit´e de Lyon, F-69622 Lyon, France: Universit´e Lyon 1, Villeurbanne CNRS/IN2P3,UMR5822, Institut de Physique des 2 Infinis de Lyon Department of Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, SouthAfrica
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We present a minimal model of asymptotic grand unification based on an SU ( ) model in a compact S /( Z × Z ′ ) orbifold. The gauge couplings run to a unifiedfixed point in the UV, without supersymmetry. By construction, fermions are embeddedin different SU ( ) bulk fields. As a consequence, baryon number is conserved, thus pre-venting proton decay, and the lightest Kaluza-Klein tier consists of new states that cannotdecay into standard model ones. We show that the top Yukawa coupling also runs to anUV fixed point with values consistent with measurements only for exactly three bulk gen-erations . Finally, the lightest massive state can play the role of Dark Matter, produced viabaryogenesis, for a Kaluza-Klein mass of about 2 . a r X i v : . [ h e p - t h ] D ec ontents The idea of unification has been employed several times in particle physics when seek-ing order in the zoo of particles and their interactions. The first famous example isthe unification of electromagnetism and the weak force within the semi-simple gaugegroup SU ( ) L × U ( ) Y [1], which then became the core of the Standard Model (SM)[2, 3]. Later, quarks and leptons were unified within the Pati-Salam gauge symmetry SU ( ) × SU ( ) L × SU ( ) R [4], which also features a left-right symmetric structure. Ulti-mately, Grand Unified Theories (GUTs) have the ambition of unifying all the forces (exceptgravity) into a unique simple gauge group (see, for example, [5–8]). The first realistic ex-ample was provided by the Georgi-Glashow theory [5], based on an SU ( ) gauge symmetry.In fact, the most minimal chiral set of fields can consist of a and a ¯5 representations,which allows for the accommodation a single complete family of the SM, without any gaugeanomaly.It is usual, in GUT model building, to assume that the unification of gauge couplingsoccurs at a specific high scale, where the low energy couplings meet via the renormalizationgroup running. In the SM, exact unification does not occur, as the gauge couplings tend– 1 –o similar values in a region of energies around 10 to 10 GeV. It is well known thatexact unification can only be achieved by enlarging the content of the SM fields, usuallythanks to supersymmetric extensions [9, 10] or by adding heavy fields [11, 12]. Whatdrives unification is the presence of incomplete multiplets of the GUT gauge symmetry,which contributes to the differential running among the three SM gauge couplings [11].In this work we consider a different kind of unification, one where the couplings unifyasymptotically. Thus, instead of crossing at a fixed energy scale, they tend to the samevalue in the deep Ultra-Violet (UV). Contrary to the standard lore, asymptotic unification isdriven by the contributions of complete multiplets of the GUT symmetry. This requirementis met in models where a compact extra dimension becomes relevant at scales higher thanthe electroweak (EW) scale and where the gauge symmetry in the bulk is unified. Thecontribution of the Kaluza-Klein states to the running, therefore, drives the gauge couplingsto a unified value at high energies [13]. While the four-dimensional gauge couplings runto zero, the five-dimensional gauge couplings run towards a fixed point in the UV [14].This mechanism has been used to define renormalizable theories in five dimensions (5D)[15] and gauge-Higgs unification models [16]. The asymptotic unification is more naturalthan the more traditional one in extra dimensional models: in fact, at energies above theinverse radius of the compact dimension, the theory approaches genuinely 5D dynamics,where a single gauge coupling is present. The deeper in the UV we probe the theory,the more closely it will behave like a unified theory. Extra-dimensional GUTs [17–20],in the traditional sense, have been considered both in 5D [21–29] and 6D [30–34], oftenaccompanied by supersymmetry to achieve exact unification, while the idea of asymptoticunification has only been pioneered in Ref. [13], without a detailed model being put forward.Here, we present the first asymptotic GUT (aGUT) based on an SU ( ) model in aflat S / Z × Z ′ orbifold. The choice of the SU ( ) group structure stems from the Georgi-Glashow model [5], where we have the smallest simple Lie group that contains the SMone. Historically, this group structure allowed for a reinterpretation of several of the fieldsas being different states of a single multiplet. The enticing structure of all known matterfields (fermions) fitting perfectly into three copies of the smallest group representations of SU ( ) , and having the correct quantized charges, became a crucial reason for people tobelieve that GUTs may be realized in Nature. We recall that the ¯5 representation containsthe charge-conjugate of the right-handed down-type quark and the left-handed lepton iso-spin doublet, while the contains the left-handed down-type quark iso-spin doublet, thecharge-conjugate of the right-handed up-type quark and of the charged lepton. Thus, asingle SM generation can be introduced in an anomaly-free way as a ¯5 + . In our extra-dimensional SU ( ) model, the chiral SM fermions are identified with the zero modes of thebulk fields. As such, it is not possible to include one whole SM generation in a single set of SU ( ) representations. This is due to the orbifold parity that breaks the gauge symmetry.Thus, for each SM family, we need to introduce a set of bulk fields in the representations , ¯5 , and ¯10 . The SM zero modes, therefore, arise from different SU ( ) multiplets. Themain consequence of this set-up is that baryon number can be preserved, thus avoiding thestrong constraints on the GUT scale from proton decay, similarly to the model in Ref. [35].The lightest Kaluza-Klein (KK) tier is made of new particles that have non-SM quan-– 2 –um numbers, which we baptize Indalo -states, from the Zulu word “indalo”, meaning cre-ation (or nature). Curiously, the same name is shared by the symbol (cid:4) (cid:109) −∩ found in prehistoriccaves near Almer´ıa in Andalusia, Spain. As we shall see in the following, the (cid:4) (cid:109) −∩ -particles areneither leptons nor quarks and have interesting properties forbidding, for example, protondecay. As the SU ( ) gauge symmetry is broken mainly via the orbifold projection, thescalar sector consists only of a scalar multiplet, containing the Higgs doublet and its (cid:4) (cid:109) −∩ -partner (a QCD color triplet Higgs H ). By construction, the new (cid:4) (cid:109) −∩ -particles have unusualbaryon and lepton numbers that forbid their decays into pure SM final states. Thus, thelightest (cid:4) (cid:109) −∩ -state, corresponding to a neutrino partner, is stable and may constitute the DarkMatter relic density.Neutrinos are also an important part of any model building beyond the SM, thus weshall also consider the effect of adding either Dirac or Majorana neutrinos to this model.Recall that in the traditional SU ( ) GUT models, the right-handed neutrinos are singletsof SU ( ) , which implies that their mass is not forbidden by any symmetry.The manuscript is organized as follows: In section 2 we introduce the model and theorbifold projections giving rise to the desired spectrum. We also show how Baryon (andLepton) number conservation arises. In section 3 we demonstrate the asymptotic unificationof the SM gauge couplings using the renormalization group equations. Furthermore, insection 4 we discuss the running of the top Yukawa couplings in relation to the presence ofan UV fixed point. In section 5 we introduce neutrino masses, while in section 6 we discussphenomenological aspects of the model, in particular baryogenesis and Indalogenesis as apotential Dark Matter generation mechanism. Finally, we offer our conclusions in section 7. We consider here the minimal aGUT model, as outlined in the introduction. It is based onan SU ( ) gauge symmetry in the bulk, broken down to the SM gauge symmetry, SU ( ) c × SU ( ) L × U ( ) Y , via orbifold boundary conditions. The background corresponds to asingle extra-dimension compactified on an orbifold S /( Z × Z ′ ) of radius R . The gaugebreaking is achieved by use of two parities, defined by two matrices P and P in the gaugespace, which correspond to a mirror symmetry around the fixed points y = y = πR / SU ( ) is broken by P at y = y = πR /
2. Whilst this is not a unique construction to achievethe aGUT scenario, it is the simplest and most minimal realisation. To further developthis model we shall first explore the orbifold parities in section 2.1, before introducing theYukawa sector of our model and, finally, the complete Lagrangian in section 2.2.
The two parities acting on the fields are associated to two matrices in the gauge space, P and P , corresponding to the two fixed points y = y = πR / A aM , where M is the 5D Lorentz index, this implies the following relations [36]: ( P ) ⇒ { A aµ ( x, − y ) = P A aµ ( x, y ) P † ,A ay ( x, − y ) = − P A ay ( x, y ) P † , (2.1)– 3 – P ) ⇒ { A aµ ( x, πR − y ) = P A aµ ( x, y ) P † ,A ay ( x, πR − y ) = − P A ay ( x, y ) P † , (2.2)where A y is the polarization along the fifth compact dimension, and the fields are periodicover y → y + πR . The radius R defines the mass scale of the KK modes of each field. Inorder to have the desired boundary conditions, we choose the P and P matrices to bediagonal with elements: P = ( + + + − − ) , (2.3) P = ( + + + + + ) . (2.4)With this choice, the bulk SU ( ) is broken to the SM group on the y = φ , transforming as a fundamental of SU ( ) , with parities: ( P ) ⇒ φ ( x, − y ) = − P φ ( x, y ) , (2.5) ( P ) ⇒ φ ( x, πR − y ) = + P φ ( x, y ) . (2.6)As in standard SU ( ) GUT models, the Higgs, φ h , is accompanied by a QCD-triplet scalar H : φ = ( Hφ h ) , (2.7)where H has no zero mode due to the Dirichlet boundary conditions on the y = SU ( ) GUTs, one SM fermion generation is usually embedded into a chiral set of ¯5 + . In the 5D construction, however, due to the boundary conditions on y = SU ( ) , it is not possible to embed a complete set of SM fermions with zero modesin a single field . Therefore our model set-up requires a doubling of the number of bulkfields as follows: ψ L / R = ( bL c ) L / R , ψ L / R = ( B c l ) L / R , (2.8) ψ L / R = √ ( T c q T c ) L / R , (2.9) ψ L / R = √ ( t Q c τ ) L / R , (2.10) In Ref. [37], the possibility of using universal boundary conditions has been used in the context ofcomposite Higgs models. This choice, however, goes beyond the orbifold construction we use here. – 4 –here the capitalized letters indicate fields that do not have a zero mode, and the super-script c indicates the 4D charge conjugate. The parities are chosen as follows: ( P ) ⇒ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ψ ( x, − y ) = + P γ ψ ( x, y ) ,ψ ( x, − y ) = + P † γ ψ ( x, y ) ,ψ ( x, − y ) = + P γ ψ ( x, y ) P T ,ψ ( x, − y ) = + P † γ ψ ( x, y ) P ∗ ; (2.11) ( P ) ⇒ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ψ ( x, πR − y ) = + P γ ψ ( x, y ) ,ψ ( x, πR − y ) = − P † γ ψ ( x, y ) ,ψ ( x, πR − y ) = − P γ ψ ( x, y ) P T ,ψ ( x, πR − y ) = + P † γ ψ ( x, y ) P ∗ . (2.12)In the above equations γ is the Dirac matrix in the (diagonal) Weyl representation. Thequantum numbers of all the components are reported in Table 1.Field ( Z , Z ′ ) SM zero mode? KK mass l ( + , + ) ( , , − / ) √ / RL ( + , − ) − / Rτ ( − , − ) ( , , − ) √ / R T ( − , + ) − / Rq ( + , + ) ( , , / ) √ / RQ ( + , − ) − / Rt ( − , − ) ( , , / ) √ / RT ( − , + ) − / Rb ( − , − ) ( , , − / ) √ / RB ( − , + ) − / Rφ h ( + , + ) ( , , / ) √ / RH ( − , + ) ( , , − / ) − / RB µ ( + , + ) ( , , ) √ / RW aµ ( , , ) G iµ ( , , ) A µX ( − , + ) ( , , − / ) − / R Table 1 . Quantum numbers and parities of all the fields (for fermions, we indicate the parities ofthe left-handed chiralities). The last two columns indicate the presence of a zero mode, and themass of the lightest KK mode.
The above-mentioned fields in five dimensions can be decomposed into towers of KKmodes, whose characteristics depend on the parities under the Z and Z ′ , which we denote ( ± , ± ) (for fermions we always denote the parities of the left-handed chirality). Of the fourcombinations, only ( + , + ) features a zero mode ( ( − , − ) for the right-handed chirality ofthe fermions), which can be associated to a SM field. In the last two columns of Table 1we indicate the presence of a zero mode and the mass of the lightest KK mode.– 5 –he lightest tier of KK modes, with mass m KK = / R , is populated by the fields thatdo not have a zero mode: a complete copy of the SM fermion families, the QCD-chargedHiggs H and the SU ( ) vector lepto-quarks A µX . As we will see at the end of the section,these fields play a special role in this model, thus we collectively name them “Indalo”( (cid:4) (cid:109) −∩ -states). As we have seen above, due to the SU ( ) breaking parity at the y = L SU ( ) = − F ( a ) MN F ( a ) MN − ξ ( ∂ µ A µ − ξ∂ A y ) + iψ / Dψ + iψ / Dψ + iψ / Dψ + iψ / Dψ − (√ Y τ ψ ψ φ ∗ + √ Y b ψ ψ φ ∗ + Y t (cid:15) ψ ψ φ + h.c. ) + ∣ D M φ ∣ − V ( φ ) , (2.13)where D M = ∂ M - ig a T a A aM with T a being the SU ( ) generators in the appropriaterepresentation, and (cid:15) is the 5-dimensional Levi-Civita symbol on the gauge indices. Theterm V ( φ ) represents a generic potential for the scalar field, which is responsible forgenerating a vacuum expectation value for the Higgs zero mode, like in the SM. We alsorecall that gauge and Yukawa couplings, via naive dimensional analysis in 5D, have scalingdimension [ m ] − / .The normalization of the Yukawa couplings is chosen to reproduce the SM ones for thezero modes. Expanding the SU ( ) multiplets into their components, the Yukawa couplingscontain the following terms: √ ψ ψ φ ∗ = bφ ∗ h q − L c H ∗ q − L c φ ∗ h T c + (cid:15) bH ∗ T c , (2.14) √ ψ ψ φ ∗ = − τ φ ∗ h l + Q c H ∗ l − Q c φ ∗ h B c + (cid:15) tH ∗ B c , (2.15)12 (cid:15) ψ ψ φ = tφ h q − tH T c − τ HT c + Q c φ h T c + (cid:15) Q c Hq , (2.16)where (cid:15) is the Levi-Civita symbol contracting the QCD SU ( ) indices. We can see fromthe above equations that each Yukawa coupling contains only one SM-like coupling be-tween zero modes. Once normalised to dimensionless 4D couplings, y f = Y f /√ πR , theycorrespond directly to the SM Yukawa couplings, and can be extended to the full flavourstructure. The only flavour violation source in this model is the SM CKM matrix, thusavoiding strong flavour bounds [39, 40]. – 6 – .3 Baryon and Lepton number conservation In SU ( ) models, strong constraints usually arise from proton decay considerations, viabaryon violating couplings of the QCD-charged scalar H [5, 41]. However, the violation ofbaryon and lepton number occurs because the SM fields are embedded in the same SU ( ) multiplet, while in our model they are not by construction, due to the specific structure in5D. From the Lagrangian in Eq. (2.13), we see that we can assign two independent globalcharges to the four fermion and one scalar fields (due to the presence of 3 independentYukawa couplings). Combined with the hypercharge, which is embedded in SU ( ) , wefind two linear combinations of the three U ( ) ’s that are preserved by the Higgs vacuumexpectation value. Their assignments on the multiplet components are listed in Table 2.For the SM fields, containing the zero modes, we can choose the charges to match thestandard baryon ( B ) and lepton ( L ) numbers. This ensures that no proton decay is allowedin our aGUT model, and the unification (compactification) scale can be potentially placedat a low scale compared to the standard GUTs. Furthermore, all (cid:4) (cid:109) −∩ -states carry both B and L charges, in values that are half of the SM unit charges. This implies that it is notpossible for them to decay into SM fields only. This property makes the lightest (cid:4) (cid:109) −∩ -statestable, and potentially a candidate for Dark Matter, as we will explore in section 5. In traditional GUT model building, the gauge couplings are supposed to run up to the samevalue at a given high scale. Thus, the most relevant feature of the renormalization groupequations (RGEs) is the relative evolution of the couplings. This is provided by the gaugebosons and the Higgs, which come in incomplete multiplets of the GUT SU ( ) symmetry[42]. The same approach has been considered in GUT models in extra dimensions [19–22].Here we will consider an antithetic scenario where it is the SU ( ) –invariant running,provided by the bulk KK modes, that drives unification of the gauge couplings asymptoti-cally at high scales. This possibility was first noted in Ref. [13], and applied to gauge-Higgsunification models in Refs [15, 16, 43, 44]. Note that the RGEs for the gauge couplingsare SM-like up to the compactification scale 1 / R , where the effect of the KK states enters.As in standard SU ( ) models, we will follow the evolution of the SU ( ) × SU ( ) × U ( ) couplings g i = { g , g , g } , where the hypercharge coupling is normalized as g = √ g ′ .The RGEs can be written as [45]:2 π dα i dt = b SM i α i + ( S ( t ) − ) b α i , (3.1)where α i = g i / π , t = ln ( µ / m Z ) , m Z is the Z boson mass, and the SM coefficients read ( b SM1 , b
SM2 , b
SM3 ) = ( / , − / , − ) . The second term includes the contributions of theKK states, in a continuum approximation, contained in the function S ( t ) = { µR = m Z Re t for µ > / R , m Z < µ < / R . (3.2)– 7 –ultiplets Fields L B Q Q ψ B cR τ L ν L ψ b R T cL -1/2 1/2 1 1 N cL -1/2 1/2 0 -1 ψ T cR T cR -1/2 1/2 1 0 t L b L ψ t R τ R T cL B cL φ H φ + φ A X X Y Table 2 . Baryon and lepton numbers for the components of the SU ( ) multiplets. We also indicatetheir electromagnetic charge Q and weak iso-spin Q . As the KK modes tend to appear in complete multiplets of SU ( ) (more precisely, addingthe KK states of adjacent even and odd KK tiers), all gauge couplings receive the samebeta function, given by b = − + n g , (3.3)where n g is the number of fermion generations in the bulk. For 3 families, as we willconsider in the following, we find b SU ( ) = − /
3. Note that this value allows for the 5Dtheory to have an UV fixed point [14, 15], a fact that will play a crucial role for theasymptotic unification.Using the RGEs for 3 families of fermions, we show in Fig. 1 the one-loop evolution ofthe three gauge couplings in terms of˜ α i = { α i ( t ) for µ < / R ,α i ( t ) S ( t ) for µ > / R . (3.4)At scales above the compactification scale, we consider an effective ’t Hooft coupling whichtakes into account the number of KK modes below the energy scale µ . We start the runningat the Z mass with the SM values { g , g , g } = { . , . , . } , while the matching to the– 8 –D running takes place at the scale 1 / R , indicated by the point where the running changessharply. Because of the absence of baryon and lepton number violation, this scale can below, so we choose 1 / R =
10 TeV as a benchmark in the plot. We can see that the couplingsnever cross, however they get very close and tend to a unified value asymptotically at highenergies. In fact, this value corresponds to the UV safe fixed point of the 5D theory. At t ≈
10, the couplings are effectively unified. This scale is well below the 5D reduced Planckmass M ∗ Pl [46], which corresponds to the largest value of t shown in the plot. Increasing1 / R does not change the picture qualitatively. α α α gauge α i Figure 1 . Running of the gauge couplings using one-loop factors, with R − =
10 TeV. The rangeof t corresponds to the Z mass ( t =
0) and the reduced 5D Planck mass.
The asymptotic behavior of the gauge couplings can be understood once the RGEs arerewritten in terms of ˜ α i at large energies. Keeping the leading terms in 1 / µR , the RGE isthe same for all gauge couplings: 2 π d ˜ α i dt = π ˜ α i + b ˜ α i . (3.5)The beta function vanishes at˜ α ∗ i ( IR ) = , ˜ α ∗ i ( U V ) = − πb , (3.6)respectively the IR and the UV fixed points. The UV fixed point only exists for b < n g ≤
3. For 4 or morebulk generations, the asymptotic unification would fail.For 3 bulk generations we find ˜ α ∗ i = π . (3.7)This value of the coupling is apparently non-perturbative. However, a more realistic as-sessment requires the estimation of the extra-dimensional loop factor, as follows: ξ ( d ) = Ω ( d )( π ) d π ˜ α , (3.8)– 9 –here Ω ( d ) is the d -dimensional solid angle. For d = ξ ( ) = ˜ α ∗ i π = π < . (3.9)We thus estimate the theory to remain perturbative at high energies, where the full 5Dnature is revealed. As already mentioned, the Yukawa couplings do not unify, as the bulk SU ( ) theory hasone independent Yukawa for each quark and charged lepton. Nevertheless, it is importantto study their running to make sure that no Landau pole develops below the 5D Planckmass.The calculation of the Yukawa running has an added difficulty compared to the gaugeone, due to the fact that gauge couplings enter in the beta function at one loop. Theissue is that this concerns not only the SM gauge couplings, but also the couplings of themassive (cid:4) (cid:109) −∩ -states A X . In principle, the running of these couplings is not linked to that ofthe gauge couplings at low energies. To take into account this uncertainty, we compute thecontribution of the KK modes under the assumption of a single 5D coupling g , and assignto it the extreme values g = g and g = g . We will consider the variation between thetwo as a systematic uncertainty in our results. Coefficients in the SM
Firstly, in the SM the three Yukawa couplings run according to the following RGEs:2 π dα t dt (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) SM = α t + α b α t + α τ α t − α α t − α α t − α α t , (4.1)2 π dα b dt (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) SM = α b + α t α b + α τ α b − α α b − α α b − α α b , (4.2)2 π dα τ dt (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) SM = α τ + α t α τ + α b α τ − α α τ − α α τ , (4.3)where α f = y f / π . These equations will be used for the running of the Yukawa couplingsbetween the EW scale and the compactification scale. Here we only consider the thirdgeneration, as they feature the largest couplings. The results can be extended to threegenerations in a straightforward way. Bulk coefficients
As explained above, we compute the coefficients of the gauge contribution to theYukawa couplings in an SU ( ) unified framework, with gauge coupling g . For the topYukawa, we obtain:2 π dα t dt = π dα t dt (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) SM + ( S ( t ) − ) [ α t + ( α b + α τ ) α t − α α t ] . (4.4)– 10 –imilarly, for the bottom and tau we find:2 π dα b dt = π dα b dt (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) SM + ( S ( t ) − ) [ α b + α t α b + α τ α b − α α b ] , (4.5)2 π dα τ dt = π dα τ dt (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) SM + ( S ( t ) − ) [ α τ + α t α τ + α b α τ − α α τ ] . (4.6)Numerical results will be expressed in terms of ˜ α f , defined as in Eq. (3.4). We focus on the running of the largest top Yukawa coupling, and neglect the effect of thesmaller bottom and tau ones. This approximation is based on the assumption that the twolatter couplings remain small at high energy, a feature that we will verify in a second step.The relevant RGE is given by Eq. (4.4), with α b = α τ = α t at high energies and keep the leading terms in 1 / µR :2 π d ˜ α t dt = π ˜ α t +
15 ˜ α t −
665 ˜ α ˜ α t . (4.7)Thus, an UV fixed point exists for˜ α ∗ t =
66 ˜ α ∗ − π = π , (4.8)where the numerical value is computed for 3 bulk generations and at 1-loop accuracy. Wecan now run down from the fixed point to the EW scale to check if the SM top Yukawacan be obtained.The results are shown in Fig. 2 for two values of 1 / R =
10 TeV and 10 TeV, and for3, 2 and 1 bulk generations. The band is due to the variation in the gauge couplings, asexplained at the beginning of this section. Remarkably, only for 3 bulk generations can theSM top value α t ( m Z ) = . ± . . < α t < .
077 for 1 / R = TeV , . < α t < .
081 for 1 / R =
10 TeV . (4.9)Recall that these values are computed at one loop. Note that values above the rangeovershoot the fixed point and thus run into a Landau pole, while smaller values will run tozero in the UV. It is remarkable that the SM value is very close to the upper bound for 3generations only. For 2 or 1 bulk generations, the SM value is always in the “Landau pole”zone. – 11 – − = TeV R − = TeV bu l k f a m ili e s α α α α t α i bu l k f a m ili e s α i bu l k f a m il y α i Figure 2 . Top down running of the top Yukawa coupling at one loop: at the reduced 5D Planckmass we fix the coupling to the fixed point value. The band indicates the uncertainty related toKK gauge couplings (see text). The largest value of t corresponds to the 5D Planck mass value.The SM value of α t at µ = m Z is indicated by the blue tick. As discussed in the previous sections, the Indalo partners of the SM particles have non-standard Baryon number assignments (see Table 2), which prevents their decay into SMstates alone. Thus, the lightest stable (cid:4) (cid:109) −∩ -state is a natural candidate for Dark Matter.The mass splitting between the lowest tier states is induced by loops [47] and by the– 12 – ψ ψ H : µ H B cR : − µ B R b R : µ b R φ + : µ + τ L : µ τ L T cL : − µ T L φ : µ ν L : µ ν L N cL : − µ N L ψ ψ A aµ T cR : − µ T R t R : µ t R W + : µ W T cR : − µ T R τ R : µ τ R X : µ X t L : µ t L T cL : − µ T L Y : µ Y b L : µ b L B cL : − µ B L Table 3 . The chemical potentials associated to the relevant fields, where the SM ones correspondto the zero modes and the Indalo ones to the lowest KK tier. effect of the Higgs. Thus, the lightest state is naturally the partner of the left-handedelectron neutrino. Being part of a doublet, its annihilation and co-annihilation cross-sections suppress the thermal relic density, which is therefore insufficient to generate thenecessary Dark Matter relic density. Furthermore, the interactions via the Z boson are toostrong to avoid exclusions by the null outcome of Direct Detection experiments [48–50].The left-handed neutrino (cid:4) (cid:109) −∩ -partner, therefore, can only constitute a minor fraction of thetotal Dark Matter relic density in the present Universe.As all the (cid:4) (cid:109) −∩ -states carry Baryon and Lepton number, a relic density can be generatedat the EW phase transition together with the Baryon asymmetry. This mechanism hasbeen used to generate an asymmetric Dark Matter relic density [51]. In our case, theDirect Detection bounds pose a strong limit on this production, thus providing conservativebounds on the radius of the extra dimension.In the next section we will discuss potential ways to allow the Indalo states to providethe total DM relic density. To estimate the relic density of (cid:4) (cid:109) −∩ -particles produced during Baryogenesis, we will rely onthe usual calculation based on the equilibrium of the chemical potential for all species ofparticles active at the time of the EW phase transition [52]. As we only want to providean estimate, we will not analyze here the dynamics of the phase transition, which is com-plicated by the extra-dimensional nature of the theory [53], and leave this study for futurework.The states we consider here, with their associated chemical potentials, are listed in Ta-ble 3, where the SM fields are associated to the zero modes and the (cid:4) (cid:109) −∩ ones to the lowest tierof the KK modes. All other states are heavier and their contributions are neglected. Fur-thermore, we assume that the three families of fermions share the same chemical potentials.The gauge and Yukawa interactions impose many relationships (recapped in Appendix A),which allow us to express all the chemical potentials in terms of 4 potentials, which wechose to be µ t L , µ W , µ H and µ . – 13 –ield Density Field Density t ( + σ t )( µ t L + µ t R ) b ( µ b L + µ b R ) ν ( µ ν L ) φ − µ φ − τ ( µ τ L + µ τ R ) h µ h T σ T ( µ T L + µ T R ) B σ B ( µ B L + µ B R ) N σ N µ N L T σ T ( µ T L + µ T R ) X σ X µ X Y σ Y µ Y H σ H µ H Table 4 . The normalised particle densities.
At the freeze-out temperature T f , the matter-antimatter asymmetry for each speciesof mass M can be written as: n = n + − n − = d dof T f µT f σ ( MT f ) , (5.1)where [54] σ ( z ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ π ∫ ∞ dxx cosh − ( √ x + z ) for fermions,64 π ∫ ∞ dxx sinh − ( √ x + z ) for bosons. (5.2)The σ function is normalised to 1 for massless fermions and to 2 for massless bosons. In thefollowing we will consider that only the (cid:4) (cid:109) −∩ -states and the top quark have a non-negligiblemass, so that the particle density will be set equal to 1 for fermions and 2 for bosons. Wehave summarised the total density of each species in Table 4. Moreover, the charges andthe iso-spins of all relevant states can be recapped in Table 2.With these ingredients we can calculate, in a straightforward way, the total baryonnumber stored separately in the SM and in the (cid:4) (cid:109) −∩ -sectors. At any given temperature, theywill depend on the total densities of each species, cf. Table 4. After the (cid:4) (cid:109) −∩ -particles exitthermal equilibrium with the SM, they will promptly decay into the lightest one, N , andrelease some baryon number to the SM sector again. For instance, we see in Table 2 thatthe (cid:4) (cid:109) −∩ -quarks T and B have baryon number − /
6, while N has baryon number − /
2, thusthe decay will release a net baryon number 1 / H , X and Y , while T has the same baryon number as N . Finally,after freeze-out, the baryon numbers in SM baryons and in N can be expressed as: B SM =
13 3 ( + σ t )( µ t L + µ t R ) +
13 9 ( µ b L + µ b R ) +
13 3 ( σ H µ H + σ X µ X + σ Y µ Y ) +
13 18 σ B ( µ B L + µ B R ) +
13 18 σ T ( µ T L + µ T R ) , (5.3)– 14 – N = − ( σ N ( µ N L ) + σ T ( µ T L + µ T R ) + ( σ H µ H + σ X µ X + σ Y µ Y ) + σ B ( µ B L + µ B R ) + σ T ( µ T L + µ T R )) . (5.4)Using the relationships among the chemical potentials, and assuming for simplicity that all (cid:4) (cid:109) −∩ -states have the same mass, i.e. σ T = σ B = σ T = σ N ≡ σ F and σ X = σ Y = σ H , we find: B SM = ( + σ t − σ F ) µ t L + ( σ F − − σ X ) µ W + ( σ X − σ F ) µ H + ( σ t − ) µ , (5.5) B N = ( σ F µ t L + ( σ X − σ F ) µ W + ( σ F − σ X ) µ H + σ F µ ) . (5.6)The mass density of N divided by the baryon density can now be expressed asΩ N Ω b = m N B N m p B SM , (5.7)where m p is the proton mass.The dependence on the chemical potentials can be further reduced by considering theEW phase transition, where we assume that it can be of 1st or 2nd order. As all (cid:4) (cid:109) −∩ -statesare vector-like, they do not contribute to the sphaleron rate, so that the same relationshipholds as in the SM [52]:3 ( µ t L + µ b L ) + µ ν L = µ t L − µ W + µ H = . (5.8)The other relevant quantities are the total electric charge and the total weak iso-spin (seeAppendix A). Under the same approximations as above, they read: Q tot = ( σ t − ) µ t L + ( + σ F + σ X ) µ W − ( + σ F + σ X ) µ H + ( + σ t + σ F − σ X ) µ , (5.9) Q tot = ( σ t − ) µ t L + ( + σ F + σ X ) µ W − σ X µ , (5.10)respectively. A first order phase transition is characterized by the vanishing of the total charge and weakiso-spin, Q tot = Q tot =
0. Together with the sphaleron condition, they allow us to write thebaryon numbers in terms of a single chemical potential. Analogous results can be obtainedfor a second order phase transition, where the vanishing of the weak iso-spin condition isreplaced by the vanishing of the Higgs chemical potential, i.e. µ = N mass and the temperatureof the phase transition. The solutions are shown in Fig. 3, where we plot the relic density asa function of the mass for three values of the EW phase transition temperature. The regionin green is excluded by the over-closure of the Universe, while the one in orange by DirectDetection via the Z interactions [55]. The current Direct Detection bounds, therefore,pushes the mass of the lightest (cid:4) (cid:109) −∩ -state above 5 to 20 TeV, depending on the temperatureof the phase transition. Future constraints, like the LUX-ZEPLIN (LZ) curve, shows we– 15 – st order phase transition 2 nd order phase transition OverclosureXENON1tLZ - - - M [ TeV ] Ω / Ω b OverclosureXENON1tLZ0.5 1 5 10 5010 - - - M [ TeV ] Figure 3 . 1 st order (left panel) and 2 nd order (right panel) phase transition results. Values of the N relic density as a function of the mass for T ∗ = v SM (solid black), v SM / v SM (dashed blue). The green shaded region is excluded by the over-closure of the Universe, while theorange shaded one is excluded by XENON1t. The dashed orange indicates the projected exclusionfrom LUX-ZEPLIN. will not push the bound significantly higher. At such a large values of the masses weobserve only minor differences between the two types of phase transitions.The strong constraints coming from direct detection requires the (cid:4) (cid:109) −∩ relic density tobe below one part in 10 compared to the baryon one. At such levels the thermal relicdensity cannot be neglected, and certainly leads to abundances above the excluded level.This analysis shows that the minimal model, as presented so far, is disfavored by cosmology.A well known way out of this problem was first used in supersymmetry [56], andconsists of generating a Majorana mass for the Dirac field, so that the two components aresplit into two Majorana mass eigenstates. However, this route is forbidden in this model bythe baryon and lepton numbers. In the following section we will show how the introductionof neutrino masses may allow the model to provide a Dark Matter candidate. The simplest way to generate neutrino masses is the introduction of a singlet field, ψ ,corresponding to the SM right-handed neutrino. For symmetry, we also introduce a secondsinglet, ψ ′ , with parities: ( P ) ⇒ { ψ ( x, − y ) = − γ ψ ( x, y ) ,ψ ′ ( x, − y ) = − γ ψ ′ ( x, y ) , (6.1) ( P ) ⇒ { ψ ( x, πR − y ) = − γ ψ ( x, y ) ,ψ ′ ( x, πR − y ) = + γ ψ ′ ( x, y ) . (6.2)Note that the second singlet allows us to combine the fields in each family into a and a ¯16 of an SO ( ) symmetry that includes the gauged SU ( ) . The parities allow us to addthe following bulk Yukawa couplings:∆ L = Y ν ¯ ψ ψ ¯5 φ + Y ′ ν ¯ ψ ′ ψ φ ∗ + h.c. . (6.3)– 16 –he field ψ contains a right-handed zero mode, which corresponds to a right-handedneutrino N R . Thus, the coupling Y ν generates a Dirac mass for the neutrinos, and isthus bound to be extremely small. Instead, ψ ′ has no zero mode, and the lightest statecorresponds to an (cid:4) (cid:109) −∩ -singlet S . The Yukawa coupling Y ′ ν ensures that S has the same baryonand lepton numbers as the (cid:4) (cid:109) −∩ -neutrino N , see Table 2.In the model extended by the two singlets, the lightest (cid:4) (cid:109) −∩ -state is S because of the lackof gauge interactions that lifts its mass. Thus, the (cid:4) (cid:109) −∩ -states produced via the Indalogenesiswill eventually decay into it via the Yukawa couplings Y ′ ν . The Dark Matter candidate S does not suffer from strong Direct Detection constraints: Fig. 3 shows that the relicdensity can be saturated for masses m S ≈ . T f = v SM .As Majorana masses are not allowed in 5 dimensions, bulk interactions can only gen-erate Dirac masses for the SM neutrinos. On the other hand, a Majorana mass for theright-handed component of ψ can be localized on either boundary. If this mass term ismuch larger than the compactification scale, a type-I see-saw can occur to suppress theMajorana masses of the left-handed neutrinos. Furthermore, this construction can generateLeptogenesis [57] at high scales, thus feeding the necessary asymmetry for the Indalogen-esis and Baryogenesis described in the previous section. We leave a detailed study of thismechanism for further investigation. We propose a concrete 5-dimensional model that realizes asymptotic Grand Unificationfor the gauge couplings. The model is based on a bulk SU ( ) gauge symmetry, brokenon a S /( Z × Z ′ ) orbifold. Due to the orbifold parity assignments, the SM fermionscannot be embedded as zero modes of a single family of + ¯5 . As a consequence, theYukawa couplings do not unify at high energies. We find that, in the minimal aGUT model,asymptotic unification is only possible for a number of bulk generations less than or equalto 3. The effective 5D gauge couplings thus run to a fixed point in the Ultra-Violet.We also studied the running of the bulk top Yukawa coupling, which also featuresan UV fixed point. The SM value of the top Yukawa at the EW scale requires exactly 3generations in the bulk. In fact, running down from the fixed point, we showed that onlyfor 3 generations the SM value of the top Yukawa is allowed. For 2 or 1 bulk generations,the top Yukawa couplings run into a Landau pole below the 5D reduced Planck mass, thusinvalidating the model.The non-unificaton of the SM Yukawas implies that baryon and lepton numbers arepreserved. Furthermore, we find that all the field components without a zero mode have B and L values which are half of those of the SM particles. For this, we suggest forthem the name Indalo ( (cid:4) (cid:109) −∩ ). The implication is two-fold: proton decay is avoided, thus thecompactification scale and unification can occur at low energies; the lightest KK state isstable and a potential Dark Matter candidate is present. A realistic scenario is achieved inpresence of an (cid:4) (cid:109) −∩ -singlet, corresponding to the right-handed neutrinos, whose relic densityis produced via Indalogenesis and Baryogenesis at the EW phase transition. We estimate– 17 –he Dark Matter mass, equal to the inverse radius, to lie in the range 1 to 6 TeV, dependingon the temperature of the phase transition, with a value of 2 . (cid:4) (cid:109) −∩ -states. As they can be singly-produced only at oneloop, the bounds from the LHC are too weak. They could be accessible at future hadroncolliders, especially in the mass range relevant for Indalo Dark Matter.In summary, we presented a minimal aGUT model in 5 dimensions, which incarnates anew asymptotic unification paradigm. The presence of UV fixed points for the bulk gaugeand top Yukawa couplings requires exactly Acknowledgements
ASC is supported in part by the National Research Foundation of South Africa (NRF)and thanks the University of Lyon 1 and IP2I for support during the collaboration visit inLyon.
A Details of the Indalogenesis
According to the interactions of the Lagrangian in Eq (2.13), the following relationshipsamong chemical potentials hold: µ H = − µ T R − µ b R = µ b L + µ N L = µ t L + µ T L = − µ B R − µ t R = µ τ L + µ T L = µ ν L + µ B L = µ τ R + µ T R = µ t R + µ T R = − µ T L − µ b L = − µ B L − µ t L , (A.1) µ = µ b L − µ b R = µ T L − µ T R = µ T R − µ T L = µ t R − µ t L = µ B L − µ B R = µ τ L − µ τ R , (A.2) µ + = µ N L − µ T R = µ t R − µ b L = µ t L − µ b R = µ T L − µ B R = µ ν L − µ τ R , (A.3)from the Yukawa couplings, and: µ W = µ T L − µ B L = µ ν L − µ τ L = µ N L − µ T L = µ t L − µ b L = µ + − µ , (A.4) µ X = µ τ L + µ B R = µ b R + µ T L = − µ t L − µ T R = µ b L + µ T R = − µ t R − µ T L = µ τ R + µ B L , (A.5) µ Y = µ ν L + µ B R = µ b R + µ N L = − µ b L − µ T R = µ t L + µ T R = − µ t R − µ B L = µ τ R + µ T L , (A.6)from the gauge vertices.The total charge density of the Universe and the weak iso-spin density are given by: Q T ot =
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