Realistic GUT Yukawa Couplings from a Random Clockwork Model

We present realistic models of flavor in SU(5) and SO(10) grand unified theories (GUTs). The models are renormalizable and do not require any exotic representations in order to accommodate the necessary GUT breaking effects in the Yukawa couplings. They are based on a simple clockwork Lagrangian whose structure is enforced with just two (one) vectorlike U(1) symmetries in the case of SU(5) and SO(10) respectively. The inter-generational hierarchies arise spontaneously from products of matrices with order one random entries.


Introduction
The puzzling flavor structure of the Standard Model (SM) has inspired a lot of model building over the past decades. The large ratios of masses and mixing parameters displayed in table 1, commonly referred to as flavor hierarchies, cannot be the result of a generic UV theory with order-one couplings.
Another feature of the SM matter sector is the peculiar structure of its five gauge representations, which strongly hints at some unified gauge group with only two SU (5) or one SO(10) representation. These so-called grand-unified theories (GUTs) in turn strongly motivate the extension of the SM to its minimally supersymmetric version, the MSSM, as the latter provides an accurate unification of the gauge couplings near a scale of the order of 10 16 GeV, commonly referred to as the GUT scale. However, the unification of couplings is much less impressive in the Yukawa sector. For instance, in SU (5) unification, one finds the GUT-scale relation Y d = Y T e for the Yukawa couplings matrices of the charged-lepton and down-quark sectors. A quick glance at table 1 (which shows the couplings at the GUT scale) reveals that this is certainly not the case. Even though supersymmetric threshold corrections are more important than in the gauge sector (and more dependent on the superpartner spectrum), it is very hard to attribute the large differences in the couplings to this alone. Typically it is possible to adjust the spectrum such that only some but not all of the Yukawa couplings become unified. Certain GUT breaking effects at the high scale thus seem to be unavoidable. Some efforts have been made to include such breaking effects, for instance via exotic Higgs representations [1], nonrenormalizable operators [2,3], vectorlike representations [4], or combinations of these [5]. In minimal SO(10) unification, all Yukawa couplings are predicted to be exactly equal, Y u = Y d = Y T e . The necessary GUT breaking effects now become even larger, as a comparison between the different columns of table 1 shows.  Table 1: Quark and Lepton data at the GUT scale in the MSSM [6]. Here, y i = y i sin β and y i = y i cos β. The values are representative, as they depend on the supersymmetric threshold corrections, and indirectly on tan β via the renormalization group running. .
The Clockwork (CW) mechanism, originally formulated in [7,8] in the context of the Relaxion, has soon been recognized as a general framework for constructing natural hierarchies [9]. In the flavor sector, it has been applied to explain the smallness of neutrino masses [10][11][12][13][14] as well as the charged flavor hierarchies mentioned above [15][16][17][18][19][20]. The CW Lagrangian is basically a one dimensional lattice ("theory space") of nearest-neighbor interactions enforced by some symmetry. The generation of hierarchies can be attributed to a controlled localization of zero modes towards the boundaries of theory space. However, it has also been pointed out [20][21][22] that when the parameters in the CW Lagrangian are chosen at random, sharp localization of the zero modes occurs in the bulk of theory space, still leading to hierarchical suppression of couplings. Moreover, when the couplings of the CW model become 3 × 3 matrices in flavor space, the three zero modes spontaneously localize at different points in the lattice, and generate inter-generational hierarchies (the vertical hierarchies in table 1) [20]. On a technical level, this is closely related to a peculiar property of products of N random matrices, which feature a very hierarchical spectrum despite their matrix elements being all of order one [16].
In this paper, we are going to build models of natural flavor hierarchies along the lines of Refs. [16,20] in supersymmetric SU (5) and SO(10) unification. The necessary GUT breaking effects are naturally present in these models, as the symmetries of the CW Lagrangian allow for renormalizable Yukawa couplings of the vectorlike fields (CW gears) with the GUT-breaking Higgs field(s). The flavor hierarchies and Yukawa unification thus have a common origin in the framework of a completely renormalizable model without any exotic Higgs or matter representations. This paper is organized as follows. In section 2 we describe our models and briefly review the mechanism of flavor hierarchies. In section 4 we perform comprehensive scans over the parameter space of these models, and quantify how well the SM flavor structure is reproduced. Some phenomenological considerations are given in section 5, and in section 6 we present our conclusions.

Charged Fermions
We denote the usual SU (5) GUT field content, consisting of 3 generations of5 and 10 each by F c and A respectively, where here and in the following we suppress all flavour indices.
Firstly, we add to this N vectorlike copies (F, F c ) i , which are charged under a vectorlike U (1) symmetry under which F i has charge i and F c i has charge −i. The index i = 1 . . . N will be referred to as a site (in theory space). In addition we introduce the U (1) breaking spurion φ with charge −1. We will denote the chiral MSSM GUT fields with an index i = 0, a notation consistent with their vanishing U (1) charge. The unique renormalizable superpotential allowed by symmetries is thus of the clockwork type where m is a mass scale, Σ is the SU (5) breaking adjoint field, and A i , B i and C i are dimensionless couplings that are 3 × 3 arbitrary complex matrices. 1 The U (1) symmetry is anomaly free (and hence can be gauged) once a conjugate field φ c is included, we will assume φ φ c such that we can simply ignore it. 2 Secondly, in a completely analogous manner we introduce copies of the fields transforming in the antisymmetric 10 representation, charged under a U (1) symmetry, with superpotential 3 Furthermore, we denote the two Higgs doublets as H c and H. The Yukawa superpotential reads with Y T = Y and M T ν = M ν . In terms of MSSM fields, this gives where the ellipsis denotes terms with the Higgs triplet. The field content of our model is summarized in table 2.
The complete superpotential of the charged fermions is W = W 5 + W 10 + W Y . Integrating out the clockwork fields with i = 0 exactly, the superpotential becomes W = W Y , while the Kähler potential turns into 1 Throughout this paper we adopt the convention that calligraphic capital letters denote complex 3 × 3 matrices. 2 Including φ c allows for the other nearest neighbor interactions, of the kind φ c F c i+1 Fi. This interaction was included in the model of Ref. [16] and leads to very similar structure for the physical Yukawa couplings. 3 We use "strength-one" notation for all gauge invariants, i.e.
Matrix notation (such as T and † ) is then exclusively reserved for flavor space.

Matter Fields
Higgs Fields with and analogously for Z . Here Y is the hypercharge (with SM normalization, Y Q = 1/6). Notice that the flavor matrices Z become hypercharge dependent and thus provide a source of GUT breaking. It is this effect that we want to exploit in order to separate the down quark and charged lepton Yukawa couplings. After canonical normalization, the physical Yukawa couplings read where the Hermitian matrices E X are defined as and Here the subscripts on the Z, Z refer to hypercharge. The eigenvalues of Z (E) are always larger (smaller) than one. Assuming no further relation between couplings it is reasonable to expect that the matrices A ( ) , B ( ) , C ( ) , and Y ( ) have O(1) complex entries. As has been shown [16,20], the matrices E X , even though not containing any a priori large or small parameters, spontaneously develop strongly hierarchical spectra, i.e., their three eigenvalues satisfy We will refer to this kind of hierarchies as inter-generational hierarchies, to distinguish them from inter-species hierarchies between the different matter representations (e.g. between top and tau). In order to better understand this spontaneous generation of inter-generational hierarchies, let us define positive parameters (2.12) In the extreme limit a + b → ∞, we have and there are no inter-generational hierarchies, while in the opposite limit of a + b → 0, we obtain a product structure which spontaneously generates large inter-generational hierarchies between the three eigenvalues. This can be understood in terms of a general property of products of random O(1) matrices [16]. The parameters a and b thus interpolate between a hierarchical and nonhierarchical situation. It is noteworthy that the hierarchies in Eq. (2.14) become independent of a, b (in the sense that only the eigenvalue's overall size but not their ratios depend on them). Interestingly, even in the case a + b ∼ 1, 4 a strong hierarchy is still present, especially between the third and second generations (see also the discussion in [20]). A complementary explanation of this mechanism can be provided by the localization of the zero modes. As has also been shown in ref. [20] the zero modes of the matter fields spontaneously localize sharply around some random site in theory space, a property first pointed out in [21] in similar models. In this interpretation, the zero modes of the three generations are localized at different sites in theory space, which explains their hierarchical overlap with the site-zero fields. The parameters a and b set a bias for this localization, the larger a and b, the more the zero modes are localized towards site zero.
Moving to a basis in which the E X are diagonal, the Yukawa couplings assume the structure (Y u ) ij ∼ Q i U c j etc, familiar from Froggatt-Nielsen [23], extra-dimensional [24][25][26], or strongly coupled [27] models, and the hierarchies of masses and mixings follow in a way similar to those (see for instance Ref. [28]). The CKM angles scale as θ ij ∼ Q i / Q j (i < j) and similarly for the PMNS anlges with Q → L. The charged fermion masses on the other hand scale as Q i U i , Q i D i , and L i E i , respectively. Then, the CKM hierarchies roughly determine the Q i hierarchies. As a general rule, this saturates the hierarchies in the down quark masses, i.e., the required hierarchies in the D i are rather mild, while the hierarchies in the up quark masses require further suppression from the U i . In the lepton sector, clearly one must avoid a large hierarchy in the L i in order to keep the PMNS angles large. The charged lepton hierarchies then come mostly from the E i parameters. From these rough considerations alone, it is already quite apparent that the structure is rather SU (5) compatible (large hierarchies in the 10 sector and mild hierarchies in the5 sector). In section 4 we will indeed show that the SU (5) model works remarkably well. However, it turns out that our ideas are even partially successful in SO(10) unified models, and it is worthwhile then to develop an SO(10) version of the model, which will be done in section 3.

Neutrinos
To generate neutrino masses we will employ the seesaw mechanism [29][30][31][32][33][34]. We will assume no clockwork fields for the neutrinos, such that we have the simple superpotential where we have already discarded the Higgs triplets. Integrating out the right handed (RH) neutrinos as well as the clockwork leptons gives the Weinberg operator where m R is a scale and D is a further dimensionless order one complex (symmetric) matrix.
It is possible to extend the previous model to SO (10). The SM fields (including the right handed neutrino) unify into a spinorial 16 representation, denoted by S. To break SO (10) down to the SM one needs more than one irreducible representation. We will choose a 45 and a vectorlike (126+126). These are the lowest-dimensional representations that satisfy the following three criteria: (i) breaking of SO(10) → SM , (ii) possibility to write Yukawa couplings in the clockwork Lagrangian with 16, 16 and a GUT breaking Higgs field, (iii) possibility to write Majorana neutrino masses [32,33]. 5

Charged Fermions
Matter Fields Higgs Fields  Table 3: Field content and symmetries of the SO(10) model. All matter fields (S i , S c i ) carry an additional generation index (not shown).
We are then lead to consider a single U (1) symmetry (the combination Q − Q of the symmetries in the SU (5) model which commutes with SO (10)). The field content is given in table 3. The model is defined by along with the unique Yukawa coupling  (5)

Neutrinos
With the symmetry assignments as in table 4, there is a unique Majorana mass term for the RH neutrinos where D is another complex order one symmetric, dimensionless matrix. All other Majorana mass terms are forbidden by the nonzero U (1) charges. Considering the nonhierarchical nature of neutrino masses, this is actually very welcome, since this implies 6 It is worthwhile to point out that the first row in table 4 corresponds to the Dimopoulos-Wilczek mechanism [] which is therefore impossible to realized within our model. 7 A further parameter, the seesaw scale, only affects the neutrino sector that the hierarchical factors E ν drop out of the Weinberg operator. We could, for instance, first integrate out the clockwork gauge-singlets, and the RH neutrino fields N c 0 would obtain the usual hierarchical wave function renormalization factor. But the normalization of the RH neutrino's kinetic term is irrelevant, as they are integrated out in the see-saw mechanism.

Simulations
In this section we would like to find a sets of parameters such that the mechanism leads to a successful generation of the SM flavor structure. We distinguish two kind of model parameters. The first kind quantify some underlying physics assumption, such as the scales of symmetry breaking, or the number of vectorlike fields. The following parameters are of this type :  N, N , a, b, a ,  We will refer to these parameters as non-stochastic or deterministic. The remaining parameters are the coupling matrices A In the absence of additional structure, such as symmetries that would constrain the form of these matrices, it is natural to assume that they have order-one complex entries. Choosing them randomly from some suitable prior distribution defines an ensemble of models (with the same physics assumptions). We can then compute the distributions of physical observables (masses and mixings), for each choice of the deterministic parameters. In order to model the property "order one" for the matrix elements, we will chose flat uniform priors with | Re(A i ) kl | ≤ 1 and | Im(A i ) kl | ≤ 1 etc. Of course, the "posterior distributions" depend to some extent on the choice of priors. 8 In order to quantify the success of the mechanism, we proceed as follows. Let us define the variables where the observables O i run over the nine physical Yukawa couplings, the three CKM mixing angles θ ij , as well as the three quantities sin 2 θ ij of the PMNS matrix. 9 It turns out that the logarithms of the observables roughly follow a multi-dimensional Gaussian distribution. It is therefore useful to approximate this distribution by a Gaussian with mean and covariance taken from the exact (simulated) distribution. This defines a χ 2 function where the meansx i and covariances C ij depend on the deterministic parameters. This χ 2 function can be used to quantify how the experimental point x exp i compares to the typical 8 We have checked though that the posterior that for instance the χ 2 values reported in tables 5 and 6 are virtually identical if we substitute the uniform priors with Gaussians. 9 For the time being we will focus our attention to these 15 observables (all taken at the GUT scale). We will comment on the remaining observables (neutrino mass squared differences and CP violating phases) below. models in the ensemble. We use as experimental input the values given in table 1. 10 Instead of χ 2 (x exp i ), an equivalent but more meaningful quantity to look at is the associated "pvalue", the proportion of models that have a larger χ 2 than the experimental point, that is, which are less likely. In our context, a p -value ∼ 1 indicates that the experimental value roughly coincides with the mean of the theoretical distribution, implying that the ensemble typically features a SM-like flavor structure.
One can then optimize the deterministic parameters in order to yield larger p -values, that is, the SM point belongs to the most likely models of the ensemble (it sits near the mean of the distribution). We should make a disclaimer though to avoid misconceptions. We are not performing a usual χ 2 fit of a model to experimental data. Rather, we are optimizing the deterministic parameters such that the theoretical distributions of models have the SM point as a typical outcome. To quantify this statements we use χ 2 models (x exp i ) (and the associated p -values) as a measure.
We will take a look at the SU (5) and SO(10) cases separately.

SU (5)
We will consider two benchmark values, tan β = 40 and tan β = 10. For each pair of N, N , we can then optimize the continuous parameters a, b, a and b . However, roughly speaking, the optimal values of a, b (a , b ) only depend on N (N ) and not N (N ). Therefore, we can group the parameters as shown in table 5 and 6. A more refined tuning of the continuous parameters can lead to slightly smaller χ 2 for some values of (N, N ), but we don't believe that this adds anything to the general conclusions. We also give in figure 1 the distributions for the case N = 1, N = 5. Several features are worth pointing out.
• The p -values in tables 5 and 6 can be quite close to one, especially in the tan β = 40 case, indicating that our mechanism results in very SM-like masses and mixings.
• As evident from Fig. 1, only weak correlations between the lepton and down sectors (for instance between mu and strange masses) persist, due to the GUT breaking effects built into the clockwork Lagrangian. One easily realizes large differences such as m µ /m s ∼ 5.
• The breaking of the degeneracy between Y d and Y e comes mostly from the5 and not from the 10 sector (the fit prefers a < b and a > b ). The reason is that the hierarchies must mainly come from the 10 sector, as explained at the end of section 2.1, but the hypercharges predict larger hierarchies in E E c than in E Q , which goes in the wrong direction. Therefore, the GUT breaking terms proportional to b cannot be too large, and N = 0 (even if not strictly needed to generate the hierarchies) is crucial to get Y d = Y T e . 11 • At larger N (N ) a too strong inter-generational hierarchy can be mitigated by larger values of a, b (a , b ), see eq. (2.13).
• On the other hand for N < 2, the inter-generational hierarchies are too small, and this cannot be compensated for by going to very small a , b , which in this regime have a universal effect on all three generations (see eq. (2.14) and the discussion there).
• There exist some deviations from the Gaussian approximation for the leptonic observables sin 2 θ 12 and sin 2 θ 23 , due to the fact that their distributions peak near the upper limit sin 2 θ ij = 1. However, the true probability density at the experimental point is larger than the the one given by the Gaussian approximation, hence our estimate for the global χ 2 is conservative.  Table 5: Simulation for the SU (5) model for the scenario with tan β = 40. We display the χ 2 of the physical couplings, corresponding to 15 degrees of freedom. The parenthesis give the p -value, the proportion of models with lower probability density (larger χ 2 ) than the SM.
Before turning to the SO(10) case, let us comment on the remaining observables, namely the mass-squared differences of the neutrinos and the CP phases in the CKM and PMNS matrices. Inverted neutrino mass ordering would imply almost complete degeneracy between the heavier two neutrinos, which is difficult to realize in our models. Hence normal mass ordering is strongly preferred. In this case, neutrino masses can always be fit rather well. We have checked that enlarging the set of observables by ∆m 2 21 and ∆m 2 32 adds very little to the global χ 2 (after adjusting the see-saw scale m R ), approximately ∼ 0.25 for N = 1, and ∼ 0.5 for N = 3. However, since there are now two more degrees of freedom, the p -values are actually slightly higher than those reported in tables 5 and 6. As for the phases δ CKM and δ PMNS , their distributions are pretty much flat over the entire range [0, 2π). Since the Gaussian approximation certainly fails for these variables, it is not very meaningful to include them in the analysis above. On the other hand the flatness of the  Table 6: Simulation for the SU (5) model for the scenario with tan β = 10. We display the χ 2 of the physical couplings, corresponding to 15 degrees of freedom. The parenthesis give the p -value, the proportion of models with lower probability density (larger χ 2 ) than the SM. distribution tells us that the experimental values are neither preferred nor disfavored in our class of models.

SO(10)
The SO(10) model depends on tan α, defined in eq. (3.3), which parametrizes the relative direction of SO(10) breaking by the 45 representation. In order to assess the favoured values of tan α, let us define the quantity where X = Q, L, U c , D c , E c . The quantity a + b X controls the average localization of the zero modes of the field X: the larger a + b X , the more it is localized towards site zero, while smaller values repel the zero modes from site zero and reduce the couplings to the Higgs. Ideally, we would like b L to be large, as it reduces the hierarchy in the PMNS angles. One has that b L is the largest amongst the b X in the regime −0.8 < tan α < 0. At the same time we need to avoid to be too close to the end points of this interval, as they correspond to points where some of the Yukawa couplings matrices exactly unify, see table 4. A value that works well in this regime is tan α ∼ −0.6, which we will use as our main benchmark.At this point, one has b X = b × {0.6, 0.46, 0.34, 0.2, 0.05} (4.6) for X = L, D c , E c , Q, U c respectively. Since the a and b parameters affect all fields simultaneously, it is no longer possible (contrary to the SU (5) case) to suppress the down type and charged lepton masses by reducing a, b without for instance also reducing the top mass. We must instead resort to large values of tan β. We find that a reasonable value is tan β ∼ 50. We show in table 7 the results of optimizing the remaining parameters a and b, and in figure 2 we show the distributions in a representative case. We can make the following observations.  Table 7: Simulation for the SO(10) model, with tan α = −0.6 and tan β = 50. The χ 2 corresponds to 15 degrees of freedom. The row marked p -value gives the proportion of models with lower probability density (larger χ 2 ) than the standard model.
hierarchies are partially erased by increasing a or b, which explains why the χ 2 values do not increase again at larger N . However, they also do not improve any further for N > 7.
• Neutrinos with normal mass ordering can again be fit easily in our model. The typical increase in χ 2 is of the order of ∼ 0.3, which for two additional degrees of freedom slightly increases the p -values.

Phenomenology
There are two principle impacts on low energy observables that could be used to constrain this class of models. Firstly, as any supersymmetric GUT model, exchange of the triplet Higgs can mediate proton decay via dimension-five operators. In minimal SU (5) GUTs, precision gauge coupling unification requires a triplet Higgs mass that allows for proton decay at a rate incompatible with data [35]. In our model, there are additional vectorlike matter fields with associated threshold corrections. Notice that only mass ratios enter in the generation of the Yukawa couplings, and the overall mass scale of these new particles is a free parameter. For each model in our scan, one can in principle adjust this mass scale and the triplet mass to achieve precise gauge coupling unification, and calculate the associated proton lifetime. Such an analysis is however not independent of the solution to the doublet triplet splitting problem, and the two issues should be dealt with together. For instance, the missing partner mechanism [36][37][38] typically require extended Higgs sectors. Even though we have presented our mechanism for a minimal Higgs sector, we expect it to work similarly well for extended Higgs sectors, as long as we can write some Yukawa interactions of the CW matter fields with the GUT-breaking Higgs fields. On the other hand it would also be interesting to try and take advantage of the CW mechanism itself to split the doublet and triplet masses. A fully realistic model in this regard, including doublet-triplet splitting and an analysis of proton decay, is left to future work.
Secondly, we should comment about low energy flavor violating signatures of these models. The wave function renormalization factors, see e.g. eq. (2.5), will also strongly Figure 2: The distribution of masses and mixings in the SO(10) model (parameters x i ), for the case N = 6. In the lower triangle, we display scatter plots of the two-dimensional marginal distributions, with the solid (dashed) contour representing one (two) sigma, and the red dots the experimental value. In the diagonal, we show the one dimensional marginal distributions over a 3 sigma range with the red lines indicating the experimental value. The numbers on the bottom of each entry are the mean and standard deviation. In the upper triangle, we show the correlation coefficients. reduce flavor violation in the soft masses [39,40]. The effect is virtually identical to strongly coupled [27] or extra-dimensional [41,42] models of supersymmetric flavor, though quite different from models with horizontal symmetries, which are more constrained than the former [40]. One of the most constraining observables is the decay of µ → eγ. If the charged lepton hierarchy is mostly coming from the E e (as in our SU (5) scenario), one has [40] A 0 100 GeV 400 GeṼ m 4 < 0.4 , (5.1) where A 0 is the trilinear soft term andm the slepton mass (similar bounds have been derived in ref. [27]). When E is somewhat hierarchical (as in our SO(10) model), the bounds become weaker. In the quark sector, the strongest bounds come from the neutron EDM which require squark massesm q 1 TeV [40]. Since the sfermion masses are dominated by the gaugino loops, squark and slepton masses are related asm q ∼ 5m , and the leptonic bounds are more constraining.

Conclusions
We have presented a renormalizable GUT model of flavor which accounts very well for the observed hierarchies of masses and mixings in the charged fermion sector. The model features one and two spontaneously broken U (1) symmetries in the case of SU (5) and SO(10) respectively. Contrary to Froggatt-Nielsen type models, the MSSM chiral matter fields are uncharged under this symmetry. We have taken advantage of the GUT breaking terms present in the most general renormalizable CW Lagrangian in order to lift the degeneracy of the down and lepton Yukawa couplings. Inter-generational hierarchies result spontaneously from products of O(1) matrices, while inter-species hierarchies can either arise from a CW-like suppression or from large tan β.
In SU (5), for a GUT breaking scale slightly smaller than the U (1) breaking scales we obtain distributions of models that feature the SM point amongst the ∼ 5% most likely models, i.e., very close to the mean value of the distribution. This requires roughly about one vectorlike copy of the5 matter fields, and 5 copies of the 10. For the best SO(10) case, the SM fits slightly worse in the distributions, belonging only to about the 60% most likely models, approximately one sigma away from the mean of the distribution. A good fit requires about N 5 vectorlike copies of the entire MSSM matter content.
The fact that the probability density is unsuppressed at the SM point can also be interpreted in terms of fine-tuning. Accidental cancellations occur only very rarely at random. We could exclude such points from our distributions, but this would at most affect the far tails of the distributions, implying that there are many non-fine tuned models which reproduce the experimental data of table 1 precisely.