Asymptotic Ultraviolet-safe Unification of Gauge and Yukawa Couplings: The exceptional case

The ultimate dream of unification models consists in combining both gauge and Yukawa couplings into one unified coupling. This is achieved by using a supersymmetric exceptional ${\rm E}_6$ gauge symmetry together with asymptotic unification in compact five-dimensional space-time. The ultraviolet fixed point requires exactly three fermion generations: one in the bulk, and the two light ones localised on the ${\rm SO}(10)$ boundary in order to cancel gauge anomalies. A second option allows to preserve baryon number and to lower the compactification scale down to the typical scales of the intermediate Pati-Salam gauge theory.

Coupling constants quantify the strength of interactions among particles and evolve with the energy of the collisions via the renormalisation group [1,2].The term unification usually indicates the crossing of the gauge couplings at a specific energy scale, assuming that a new larger gauge symmetry emerges after that point.This mechanism yields a unified description of the electroweak and strong interactions as a single force, within the socalled Grand Unified Theories (GUTs) [3][4][5].GUTs feature the assembling of matter as well as gauge degrees of freedom within multiplets in a rather elegant and dangerous way.In fact, gauge unification typically implies relations among the Standard Model (SM) couplings and the presence of new gauge states, often bringing embarrassing predictions, such as fast proton decay [6].While countermeasures to these shortcomings are well-known [6,7], they typically tend to partially obscure the beauty and the simplicity of the original models: the unification scale is pushed close to the Planck mass and fields in large representations of the unified group are often a must [8].The latter leads to an unresolved issue: GUTs, especially based on supersymmetry, feature Landau poles for the unified coupling evolution soon after unification, hence maiming the validity of the theory.
The idea of asymptotic unification solves this problem [9], as it is based on the requirement that the theory should flow to a non-trivial ultraviolet (UV) fixed point at high energy, without the need of a specific unification scale.This is possible in models with a single extra space dimension [10,11], which yields a power running of the gauge couplings [12,13].The extra dimension is compactified on an interval, where the bulk could be flat or warped [14] as the high-energy behaviour is the same.The model features a five-dimensional unified gauge symmetry in the bulk, so that at high energies (small distances) the unified behaviour emerges.The bulk gauge group is broken by boundary conditions, and the SM fields are identified with the zero modes of the bulk fields, implying a completely different arrangement of matter and gauge fields within multiplets as compared to traditional unification.
Extra-dimensional GUTs were widely explored [15][16][17][18][19], where the unification is due to a sub-leading logarithmic running of the gauge couplings [20].Asymptotic unification, instead, is driven by the contributions of complete multiplets, not by incomplete ones as in the standard GUT.Hence, it is the contribution of the bulk resonances that drives it [12,21].Asymptotic unification is a more natural choice than the standard one in extra dimensional models: in fact, at energies above the inverse radius of the compact dimension, the theory approaches the extra-dimensional evolution, showing that the electroweak and strong interactions have always been a single force at small distances.They appear different at low energies only due to the breaking effect of the compactification.A concrete model of asymptotic Grand Unification (aGUT) was recently proposed, based on the minimal SU(5) gauge group [22, 23][24].Interestingly, the running of the Yukawa couplings imposes critical conditions on these models, for instance ruling out the traditional SO (10) symmetry [25].
A more ambitious programme would require the unification of all the couplings of the SM, namely gauge and Yukawa couplings, at least for one generation.In extra dimensions, this has been explored in the context of gauge-Higgs unification models [26].Within the asymptotic unification paradigm, this can be achieved if the UV fixed points for all the couplings coincide.Supersymmetry comes in handy, as it naturally relates couplings of bosons (gauge bosons) to those of fermions (gauginos).Within this framework, the Lie group E 6 [27,28] offers an exceptional opportunity, as the SM fermions can be embedded within the adjoint representation, hence stemming from E 6 gauginos [29].As the Yukawa couplings are generated from the E 6 gauge interactions, they share the same fixed point in the UV.The model, therefore, consists of a supersymmetric E 6 gauge theory in five dimensions (5D) with matter fields in the fundamental 27.The fifth dimension is compactified on the orbifold S 1 /Z 2 × Z 2 .From the four-dimensional (4D) point of view, this theory has N = 2 supersymmetry [30], hence the gauge and matter superfields consist of Gauge : Matter : where W α is the vector superfield, while Φ are the chiral superfields, with Φ c 27 having conjugate quantum numbers as compared to Φ 27 .The model Lagrangian can be constructed in terms of these components [30][31][32].The orbifold projection, encoded in the two parities Z 2 and Z 2 centred on the two endpoints of the interval, break both the N = 2 supersymmetry to N = 1 in 4D and the gauge symmetry.For the latter, there are three possible patterns [33]: C: where the subscript "L" and "R" refer to the custodial symmetry in the SM.Choosing any pair of the above patterns for the two orbifold projections leads to a 4D gauge theory based on Pati-Salam (PS) [34] times an additional abelian U(1) symmetry: SU(4) × SU(2) L × SU(2) R ×U(1) ψ .However, only one choice is phenomenologically viable.For the combination B-C, it is not possible to obtain chiral zero modes for the SM fermions, while for A-C the zero mode spectrum does not allow for the breaking of the PS symmetry.Hence, the only viable model is based on A-B as illustrated in Fig. 1.The field content is summarised in Fig. 2, where we highlight the decomposition with respect to the various subgroups of E 6 , and we include two matter fields in the fundamental representation, 27 and 27 , with different orbifold parities.As we will see below this is the minimal and unique embedding of one SM generation in the bulk.
A similar set-up was first proposed as a string-inspired standard GUT [29].Our proposal differs in two crucial features: a) the unification is driven asymptotically by the UV fixed point; b) all non-SM zero mode fields receive mass without the need for additional bulk fields.The second feature is crucial to maintain the UV fixed point of the theory.
The UV fixed point emerges from the power law corrections to the renormalisation group evolution of the effective 4D gauge coupling [12].The beta function can be expressed as an effective 5D 't Hooft coupling [21], defined in terms of the 4D coupling α = g 2 /4π as where µ is the renormalisation scale and m KK is the mass of the first Kaluza-Klein (KK) state.At one loop, the beta function for α reads which is valid for µ m KK and has a UV zero at α * UV = 2π/b 5 for b 5 > 0. For our model, we find [35] where C(G) = 12 and T (27) = 3 for E 6 [36].Hence, only one more fundamental can be added in the bulk before the UV fixed point disappears.As a consequence, the model in Ref. [29] does not have such UV fixed point.In our model, instead, the UV fixed point remains perturbative, hence endorsing the model with enhanced calculability.We remark that this behaviour remains the same for both flat and warped extra dimensions [37].
From the 4D point of view, the theory is invariant under PS×U(1) ψ , which we use to classify the relevant states.At the zero mode level, besides the gauge superfields in W α 78 , the chiral superfields contain the following states: shown as coloured symbols in Fig. 2. We can already see that the 27 contains the left-handed SM fermions (4, 2, 1) 1 and two Higgs doublets in (1, 2, 2) 2 .The PS Yukawa couplings stem from the gauge interactions of the 27 with the Φ 78 component of the gauge supermultiplet [38,39], leading to: thus the right-handed SM fermions must be the ( 4, 1, 2) −3 component of the gauge Φ 78 .The gauge interactions of the 27 contain the following zero-mode terms: The importance of the couplings above is related to the breaking of the PS×U(1) ψ gauge group down to the SM one.In fact, the ( At the UV fixed point, the matching between E 6 and the PS×U(1) ψ couplings reads: For the SM Yukawa couplings in Eq. ( 12), we have where y ↑ = y t = y ντ is the Yukawa of up-type fermions, while y ↓ = y b = y τ for down-type ones.The identification of up and down-type Yukawas occurs at the PSbreaking scale, which is typically close to m KK , while the relation between top and bottom mass also depends on the ratio of Higgs vacuum expectation values, as typical in supersymmetric models [42].This ratio, expressed in terms of tan β, requires where the masses are evaluated at the KK scale.In Fig. 3 we show a schematic plot of the renormalisation group evolution of the SM couplings in the E 6 model.For simplicity, we identify the PS and U(1) ψ breaking scales to m KK , and fix the supersymmetry breaking scale to 10 TeV, above which the minimal supersymmetric SM (MSSM) is a good description.The couplings correspond to the usual 4D ones up to the scale m KK , above which they are replaced by the corresponding 5D 't Hooft couplings, defined in Eq. ( 4).Also, we plot the couplings rescaled to the E 6 values, as in Eqs. ( 14) and ( 15), while the usual PS matching is applied above m KK .This plot clearly demonstrates that the gauge and Yukawa couplings of the third generation do unify to a single value thanks to the UV fixed point, independently on the value of m KK .However, due to the constrained bulk structure, the light generations must be localised on one of the two boundaries.Before addressing this issue, there are two related features of the bulk interactions: baryon number conservation and the cancellation of 4D gauge anomalies.
Regarding the former, we recall that the theory, at the level of the SM gauge invariance, features five U(1) symmetries.Besides the three gauged ones, U(1) B-L ⊃ SU(4), U(1) R ⊃ SU(2) R and U(1) ψ , there are two global charges associated to the matter fields, U(1) 27 and U(1) 27 .Among the first three, the SM hypercharge is defined as usual in PS models as There is a single global charge that remains after the breaking of PS and U(1) ψ , and that is not carried by the Higgs for Yukawas and α1 = 5 3 α for hypercharge, where αR comes from the usual PS matching. fields: where the global charges are normalised to unity.On the SM fields, this charge matches baryon number.Being respected by all bulk interactions, it protects the proton from decaying.As in the minimal SU(5) aGUT model [22], the components of the bulk fields with opposite parities on the two boundaries, shown by the black symbols in Fig. 2, have unusual Q B assignments, hence the lightest state among them is stable.All zero modes have standard baryon number assignments, including the non-SM states.A detailed description of the components is presented in the supplementary material.
Another feature of the bulk model is the presence of 4D anomalies at the level of the gauge symmetry PS×U(1) ψ , which can only stem from the matter fields as the 78 is real.As shown in Fig. 2, however, the zero modes of the 27 and 27 form effectively complete representations of SO(10) ⊃ PS, namely 16 1 + 10 2 + 1 −4 .Hence, the PS gauge symmetry is anomaly-free.For instance, the 16 1 is formed by the (4, 2, 1) 1 zero mode in Φ 27 and the ( 4, 1, 2) 1 in Φ c 27 .The 4D anomalies, therefore, only involve the U(1) ψ charges, where the coefficients respect A(16 1 ) = A(10 2 + 1 −4 ).The U(1) ψ gauge anomalies can only be cancelled by adding fields on the SO(10) boundary.This leads to two possible models: i) Two 16 −1 multiplets, whose components match the two SM light generations.Hence, the total number of SM generations is predicted to be three by gauge anomaly cancellation.However, baryon number is violated by the localised gauge and Yukawa interactions.
ii) Two 10 −2 + 1 4 , corresponding to massive states, while the light generations are localised on the other boundary.
In the first case i), the two localised superfields Φ i 16−1 allow for the following couplings to the bulk field containing the Higgs doublets: This term, and the SO(10) gauge interactions of the localised fields, violate baryon and lepton numbers, as in traditional GUT models, via couplings with the first KK resonances.For flat extra dimensions, this would lead to a direct bound m KK 10 16 GeV [43].We remark that warping the extra space can lead to a mild suppression of these coupling [44], hence it would be feasible to lower the KK scale by one or two orders of magnitude.Furthermore, additional localised superfields can be added in order to explain the different values of the Yukawa couplings, as done in minimal SO (10) GUTs [8]: the presence of large representations does not spoil the UV fixed point as they only contribute to logarithmic running, which is overpowered by the bulk power-law running.As an example, in Fig. 3 we show a case with m KK = 10 15 GeV.The mixing between the third generation and the light ones, however, is forbidden by the U(1) ψ gauge symmetry, and it requires additional localised fields, see supplementary material.
The above fields match the bulk field components containing the third generation, hence the three generations share similar localised couplings.The localised Yukawas are written in terms of the following 3 × 3 matrices where i = 3 corresponds to the bulk fields and the second term gives mass to the unwanted components.As the above couplings have the same structure of the bulk ones, the baryon number in Eq. ( 17) remains conserved and the m KK scale can be lowered compared to case i).In this model, it can be as low as the lowest scale allowed by PS to be around 2000 TeV, which is obtained by using the current limit on Br (K L → µ ± e ∓ ) < 4.7 × 10 −12 [45](see also e.g.[46][47][48]).
In conclusion, in this letter we presented a new aGUT based on a supersymmetric exceptional E 6 gauge symmetry.Its uniqueness stands in the presence of a single UV fixed point for gauge and Yukawa couplings of the third generation.The number of SM generations is predicted by gauge anomaly cancellation.We highlighted a second option, which preserves baryon number and allows to lower the compactification scale down to a few thousand TeV.This exceptional model has far reaching implications both for low energy phenomenology, for instance in the flavour sector, and at high energies, via new model building opportunities for UV completions.
[36] This result stems from a direct 5D loop computation.
More commonly, b5 is computed in terms of KK states, leading to The difference is mainly due to the phase space integration in 4D versus 5D, and it can be considered as an uncertainty on the numerical predictions.
[37] The two geometries have the same short distance behaviour.Hence, the only difference will be encoded in threshold effects near the compactification scale.[41] In terms of SO(10) × U(1) ψ components, the superpotential term reads: Supplementary material

5D supersymmetric Lagrangian
The gauge fields consist of one 5D vector V M (M = 1, . . .5), one 5D spinor (λ, λ ) and a real scalar σ, transforming as the adjoint irreducible representation (irrep) of E 6 .They can be expressed in terms of the following N = 1 superfields: where Σ = (σ+iV 5 ).Here, W α is a vector superfield, while Φ is a chiral one.Together, they form a 5D hypermultiplet, which resembles that of N = 2 supersymmetry in 4D.The 5D action can be written as: where ∇ y e 2gV = ∂ y e 2gV − g Φe 2gV − e 2gV gΦ .
A 5D matter multiplet in the irrep R consists of a Dirac spinor ψ R and two scalar states φ R1 and φ R2 .Writing the Dirac spinor in terms of 4D Weyl spinors as the fields can be organised into two chiral superfields where the second one transforms as conjugate irrep of E 6 .In other words, Φ contains the left-handed spinor, while Φ c contains the charge-conjugate right-handed one.The 5D action, including interactions with the gauge fields, can be written as The third term above takes care of the 5D Lorentz invariance by introducing a coupling between the two chiral supermultiplets.Also, the above Lagrangian has a global U(1) symmetry under which

E6 breaking and orbifold boundary conditions
In general, the components of an E 6 multiplet will receive different parity assignments.Let us consider a parity P that breaks E 6 → H, so that an irrep R of E 6 decomposes in components χ j .The parity assignments of each component χ j will read: where η R is an overall sign that can be chosen different for different fields in the irrep R, while the relative parities P j only depend on the E 6 irrep (and are the same for all fields).We consider the following breaking patterns: B : C : The intersection of any pair of them leaves the Pati-Salam group unbroken, SU(4) × SU(2) L × SU(2) R , plus U(1) ψ .
A detailed list of the bulk field components with their assigned orbifold parities can be found in Tables III, IV and V, for the choice A-B, which is presented in the main text.Two combinations are broken by the vacuum expectation values that break PS×U (1) ψ down to the SM: where we assume that the PS breaking is due to the neutrino singlets zero modes contained in the adjoint (via the Scherk-Scharz mechanism).We can further define the hypercharge as follows: Henceforth, it remains the freedom of defining two unbroken global symmetries.There is a unique charge which vanishes on both Higgs doublets, hence it will not be broken by the Higgs vacuum expectation values: where the normalisation matches the baryon number of quarks.The second combination must be non-zero on at least one of the Higgs doublets: choosing to normalise the charge to be 1 on ϕ h2 , we obtain The charge assignments are listed in Tables III, IV and V.Note that the PS symmetry may also be broken by the ( 4, 1, 2) 1 in 27 , however this leads to the same charge definition as above.

Localised superpotential on the SO(10) boundary
In the model with two localised generations on the SO(10) boundary, mixings with the bulk third generation are not possible, as they are forbidden by the Q ψ charges.As a solution, one can introduce a minimal set of localised fields, such as a set of localised Higgs states in the 10 0 , which allows for ỹi hence generating a mixing in the left-handed sector of the theory.Note that no additional gauge anomalies are generated by such localised Higgs states.A mixing between the localised and bulk Higgs fields could be generated by SUSY-breaking quartic couplings, or by introducing additional singlets charged under U(1) ψ yielding the following superpotential:

FIG. 2 .
FIG.2.Illustration of the three bulk fields in terms of PS×U ψ components of the E6 representations.The position on the grid indicates the quantum numbers under SU(2)L × SU(2)R, while the symbols represent SU(4) representations.The SO(10) components are linked by dashed lines and labelled.Finally, the colours indicate the presence of a zero mode: red for the W α 78 component and blue for the Φ78, green for the matter Φ and orange for the Φ c .The black symbols correspond to components without a zero mode.

FIG. 3 .
FIG. 3. Schematic running of the SM gauge couplings and third generation Yukawa couplings for two values of mKK, where the Yukawa values correspond to tan β = 40.The couplings are rescaled to match the E6 unification: αx = 2 y 2 x 4π

TABLE I .
Intrinsic parities for the fundamental 27 of E6.The colours label PS fields in the same irrep of H.

TABLE II .
Intrinsic parities for the adjoint 78 of E6.The colours label PS fields in the same irrep of H.At the level of the SM gauge symmetry, the model has 5 U(1) factors: U(1) B-L in SU(4), U(1) R in SU(2) R , U(1) ψ , U(1) 27 and U(1) 27 .The two last ones are global, while the first 3 are gauged.

TABLE IV .
Field decomposition for the Matter multiplet R = 27 .