Tension between neutrino masses and gauge coupling unification in natural grand unified theories

The natural grand unified theories solve various problems of the supersymmetric grand unified theory and give realistic quark and lepton mass matrices under the natural assumption that all terms allowed by the symmetry are introduced with O(1) coefficients. However, because of the natural assumption, it is difficult to achieve the gauge coupling unification without tuning, while keeping neutrino masses at realistic values. In this paper, we try to avoid this tension between the neutrino masses and the gauge coupling unification, by introducing suppression factors for several terms. These suppression factors can be understood by approximate symmetries in some of the solutions. We show that one of the most important results in the natural GUT scenario, that the nucleon decay mediated by superheavy gauge fields is enhanced due to smaller unification scale while the nucleon decay mediated by superheavy colored Higgs is suppressed, can change in those models proposed in this paper.


I. INTRODUCTION
The grand unified theory (GUT) [1] is the most promising model beyond the standard model (SM).The GUT not only unifies three of the four forces in nature except gravity, but also unifies matter such as quarks and leptons.Furthermore, the GUT already has been supported from experiments for each unification.With respect to force unification, it has been shown that by introducing supersymmetry (SUSY) [2], the three running gauge couplings in the SM coincide on the grand unified scale Λ G ∼ 2 × 10 16 GeV [3].As for the unification of matter, the various hierarchies of quark and lepton masses and mixings measured by various experiments [4] can be reasonably explained by the unification of matter in the SU (5) GUT, in which one generation quarks and leptons can be unified into 5 and 10 of SU (5).In SU (5) unification, the single assumption, that 10 fields induce stronger hierarchies in Yukawa couplings than 5 field, explains not only various mass hierarchies of quarks and leptons -the up-type quarks have the strongest mass hierarchy, the neutrinos have the weakest mass hierarchy1 , and the down-type quarks and the charged leptons have the intermediate mass hierarchies, but also, simultaneously, that the quark mixings are smaller than the lepton mixings.This is a non-trivial result, and hence this can be understood as the experimental support for the matter unification in SU (5) unification.Unfortunately, SUSY GUT has two main problems.The first is that the unification of matter also unifies the mass matrices of quarks and leptons, which are inconsistent with the observed values [? ].The second problem is that the mass of the SM Higgs partner must be sufficiently large compared to the SM Higgs mass to obtain sufficiently stable proton [5], which is difficult to achieve without fine tuning.
The natural GUTs [6][7][8] solve those SUSY GUT problems under the natural assumption that all terms allowed by the symmetry are introduced with O(1) factors.As a result, we have a natural GUT that becomes the SM at low energy.Unfortunately, the problem is that in order to achieve unification of the gauge coupling constants, many O(1) coefficients in the range of 0.5 to 2 must be artificially choosed, which in a sense is a fine tuning.As will be discussed in detail in this paper, this issue is related with the measured neutrino masses.
That is, under the natural assumptions in the natural GUT, there is a tension between the unification of gauge coupling constants and the measured neutrino masses.I. Field contents of natural SO (10) GUT with U (1) A charges. ± shows Z 2 parity.The half integer U (1) A charges play the same role as R-parity.
In this paper, we discuss how to resolve this tension.In particular, we consider the possibility of solving this problem by considering cases in which small suppression factors, rather than O(1) factors, are applied to several particular terms.After building the models, we discuss the origin of those small suppression factors, such as approximate symmetries.
Moreover, we discuss the predictions of nucleon decay in those models.Interestingly, the predictions of the natural GUT, that nucleon decay via dimension 6 operators is enhanced due to smaller unification scale and nucleon decay via dimension 5 operators is suppressed, can change in those models.
After this introduction, we review the natural GUT in section II.In section III, we discuss the solutions to solve this tension and approximate symmetries to make this solution natural.
We also build explicit natural GUT models and discuss the nucleon decay in these GUT models.Section IV is for discussions and summary.

II. NATURAL GUT AND ITS PROBLEMS
In this section, we review the SO( 10) natural GUT [6,7] and its problem, which also appears in the E 6 natural GUT [8].One of the most important features in the natural GUT is that all terms allowed by the symmetry are introduced with O(1) coefficients.Because of this feature, once we fix the symmetry of the model, the definition of the model can be done except O(1) coefficients.Under the symmetry SO(10) × U(1) A × Z 2 , typical quantum numbers of the field contents are given in Table I.In this paper, we use large characters for fields or operators and small characters for their U(1) A charges.The U(1) A [9] has gauge anomalies which are cancelled by the Green-Schwarz mechanism [10] 2 , and the Fayet-Iliopoulos (FI) term [11] ξ 2 d 2 θV A is assumed, where V A is a vector multiplet of the U(1) A .It is surprising that the various problems in SUSY GUT scenarios, including the doublet-triplet splitting problem, can be solved in this model with the above natural feature.Unfortunately, there is a tension between the neutrino masses and the unification of the gauge couplings.Let us explain them in details in the review of the natural GUT below.
A. Anomalous U (1) A gauge symmetry First, for simplicity, we consider a simpler model in which we have only three matter fields Ψ i , and two negatively charged fields H, and Θ in Table I.The superpotential invariant under U(1) A is given as where Λ and c ij are the cutoff of the model and O( 1) coefficients.If we assume that only Θ has a non-vanishing vacuum expectation value (VEV), which is determined by the D-flatness condition of U(1) A as Θ = ξ ≡ λΛ, the interaction terms in the above superpotential becomes the hierarchical Yukawa interactions as when λ < 1.In this paper, we take λ ∼ 0.22, which is approximately the Cabibbo angle.
The realization of the hierarchical structures of Yukawa couplings from higher dimensional effective interactions by developing the VEV of some fields, which breaks a (flavor) symmetry, is often called the Froggatt-Nielsen mechanism [12].The important point is that Yukawa hierarchies can be reproduced under the natural assumption that all terms allowed by the symmetry are introduced, including higher dimensional terms.However, it is quite rare to adopt this natural assumption even in the GUT Higgs sector in which the GUT group is spontaneously broken into the SM gauge group, mainly because it is difficult to control 2 Strictly, the fields in Table I alone may not satisfy the conditions for the anomaly cancellation, but the arbitrariness of the normalization of U (1) A and the introduction of a few new SO (10) singlet fields can satisfy those conditions.
infinite number of higher dimensional terms.Within the same theory, it is not reasonable for the Yukawa sector to adopt this natural assumption and the Higgs sector not.The natural GUTs are the theories in which this natural assumption is adopted even in the GUT Higgs sector as well as in the Yukawa sector.
Note that the Higgs mass term λ 2h H 2 is forbidden when h < 0 because of the holomorphic feature of the superpotential, that is called the SUSY zero mechanism, or the holomorphic zero mechanism.The SUSY zero mechanism plays important roles in controling the infinite number of higher dimensional terms and in solving the doublet-triplet (DT) splitting problem.We will explain them in the next subsection.

B. Higgs sector in natural GUT
In this subsection, we will briefly review the GUT Higgs sector in the natural GUT, which breaks SO (10) into Y and solves the DT splitting problem under the natural assumption.
One of the most important assumptions is that all positively U(1) A charged fields have vanishing VEVs.This assumption not only allows the SUSY zero mechanism to work, but also to control an infinite number of higher dimensional terms.Under this assumption, it is easy to show that the F-flatness conditions for negatively charged fields are automatically satisfied.The F-flatness conditions of positively charged fields determine the VEVs of negatively charged fields.Thus, ignoring the D-flatness conditions, if the number of positively charged fields equals the number of negatively charged fields, the VEVs of all negatively charged fields can be determined in principle.In order to break SO (10) where W X denotes the terms linear in the X field.Each W X includes finite number of terms because of the SUSY zero mechanism.Note that only finite number of terms are important to fix the VEVs although the infinite number of higher dimensional terms are introduced.
Now we discuss how to determine the VEVs by W X .First, we consider W A ′ , which is given as where the subscripts 1 and 54 denote the representation of the composite operators under the SO( 10) gauge symmetry.Unless otherwise noted, the O( 1) coefficients are omitted and we take Λ = 1 in this paper.The F-flatness condition of A ′ fixes the VEV of A. One of the 6 vacua3 becomes the Dimopoulos-Wilczek (DW) form [13] as This VEV of A plays an important role in solving the DT splitting problem.Actually, through the triplet Higgses become massive while the doublet Higgses remain massless.One pair of doublet Higgses becomes massive through the mass term λ 2h ′ H ′2 .(Note that to determine the mass spectrum, the terms which include two positively charged fields must be considered. ) Then only one pair of doublet Higgses becomes massless, and therefore, the DT splitting problem can be solved.The effective colored Higgs mass for nucleon decay becomes λ 2h which is larger than the cutoff scale because h < 0.
The VEVs of C and C, which is important to break SU(3 into G SM , are induced by the F-flatness condition of S from the superpotential Since λ 2ka A 2k ∼ 1, basically the last term in Eq. ( 6) does not change the following result.
The F-flatness condition of S gives CC ∼ λ −(c+c) , and thus the D-flatness condition of SO (10) . Note that the VEVs of C and C are determined by their charges again.The F -flatness conditions of C ′ and C′ realize the alignment of the VEVs C , C , and A , and impart masses on the pseudo Nambu-Goldstone fields. 4This mechanism proposed by Barr and Raby [14] is naturally embedded in the natural GUT.W C ′ and W C′ are given as U(1) B−L , which are obtained by decomposition of 16 of SO( 10), has non-vanishing VEV.

C. Mass spectrum of superheavy particles
Since all the interactions are determined by the symmetry, mass spectrum of superheavy particles are also fixed except O(1) coefficients in the natural GUTs.The mass spectrum are important in calculating the renormalization group equations (RGEs) for the gauge couplings.Note that we have to consider also the terms which include two positively charged fields in order to examine the mass spectrum.
The spinor 16, the vector 10 and the adjoint 45 of SO( 10) are decomposed under 10 where the quantum numbers of G SM are explicitly written as , G(8, 1) 0 , and W (1, 3) 0 .
4 Without W C ′ and W C′ , the superpotential for fixing A becomes independent of the superpotential for fixing the VEVs C , C .It means that accidental global symmetry appears and as a result, pseudo Nambu-Goldstone fields appear by breaking the global symmetry.
First, let us consider the mass spectrum of 5 and 5 of SU (5).The mass matrices M I (I = D c (H T ), L(H D )) can be written as where α L(H D ) is vanishing and Only one pair of doublet Higgs becomes massless, which comes from Next, we consider the mass matrices for 10 of SU( 5), which are given by Here, α Q and α U c are vanishing because these are Nambu-Goldstone modes, but α E c ∼ O (1).
Finally, we consider the mass spectrum for 24 of SU( 5).The mass matrices M I (I = G, W, X) are given by The mass spectrum for G and W are (λ a ′ +a , λ a ′ +a ), while for X it becomes (0, λ 2h ′ ).The massless mode of X is eaten by the Higgs mechanism.
Since all symmetry breaking scales and all the mass spectrums of superheavy particles are fixed by anomalous U(1) A charges, we can calculate the running gauge couplings and discuss the gauge coupling unification.In this paper, we study the running gauge couplings obtained by one-loop RGEs.Note that superheavy particles from the matter sector which is discussed in the next subsection are complete multiplets of SU( 5), and therefore, they do not affect the conditions for unification of the gauge coupling constants.
In the natural SO( 10) GUT, SO( 10) is broken by the VEV Now let us discuss the conditions of the gauge coupling unification where The gauge couplings at the scale Λ A are obtained by one-loop RGEs as where M SB is a SUSY breaking scale.Here, (b 1 , b 2 , b 3 ) = (33/5, 1, −3) represent the renormalization group coefficients for the minimal SUSY standard model(MSSM) and ∆b ai (a = 1, 2, 3) denote the corrections to these coefficients arising from the massive fields with mass m i , which can be read from the Table The last term in Eq. ( 17) is caused by the breaking SU(2) R × U(1) B−L → U(1) Y due to the VEV C .The gauge couplings at the SUSY breaking scale M SB can be obtained by the success of the gauge coupling unification in the MSSM as where α −1 G (Λ G ) ∼ 25 and Λ G ∼ 2 × 10 16 GeV.The above conditions for unification are rewritten as where MI are the reduced mass matrices where massless modes are omitted from the original mass matrices and rI are rank of the reduced mass matrices.In our scenario, the symmetry breaking scales Λ A ∼ λ −a , Λ C ∼ λ − 1 2 (c+c) , and the determinants of the reduced mass matrices are determined by the anomalous U(1) A charges; The unification conditions Λ ∼ Λ G .
(33) Surprisingly, the above unification conditions do not depend on the anomalous U(1) A charges except h.This can be shown to be a general result in the GUT with the anomalous U(1) A [7].It is important that the cutoff scale in the natural GUT is taken to be around the usual GUT scale.It means that the true GUT scale Λ A ≡ A ∼ λ −a Λ becomes smaller than Λ G .
Therefore, the nucleon decay via superheavy gauge field exchange is enhanced and it may be seen in near future experiments.
Unfortunately, in the natural GUT model in Table I, we take h = −3 not h = 0. Of course, to forbid the explicit SM Higgs mass term H 2 , h must be negative.But only because of that, we can take larger h, for example, h = −1.We take h = −3 in order to obtain realistic neutrino masses.In other word, if we take h ∼ 0, the neutrino masses become too small.We will explain them in the next subsection.
E. Matter sector in natural SO (10) GUT In this subsection, we will briefly review how to obtain realistic quark and lepton masses and mixings in the natural SO(10) GUT.Especially, neutrino masses will be explained in details because we will introduce a tension between the neutrino masses and gauge coupling unification condition later.
If the Yukawa interactions have been obtained only from the superpotential in Eq. ( 1), the model would be unrealistic because of the unrealistic SO( 10) GUT relations for the Yukawa couplings.The easiest way to avoid this unrealistic SO (10) GUT relations is to introduce 10 of SO( 10) as a matter field in addition to three 16 as in Table I.The model has four 5 and one 5 of SU( 5) since 16 and 10 of SO( 10) are decomposed under SU(5) as 16 = 10 + 5 + 1 and 10 = 5 + 5.One of four 5s becomes superheavy with a 5 field through the interactions obtained as Note that the higher dimensional interactions, λ ψ i +ψ j +c+c+h Ψ i Ψ j CCH and λ ψ i +ψ j +2La+h Ψ i Ψ j A 2L H, give the same order contributions to these Yukawa couplings as λ ψ i +ψ j +h Ψ i Ψ j H after developing the VEVs, CC ∼ λ −(c+c) and A ∼ λ −a .Because of this feature, the SU( 5) GUT relation Y d = Y T e can naturally be avoided in the natural GUT.Thus, we can obtain the Cabibbo-Kobayashi-Maskawa (CKM) [15] matrix as which is consistent with the experimental value if we choose λ ∼ 0.225 .
The right-handed neutrino masses are obtained from the interactions as Thus, the neutrino mass matrix is written as The Maki-Nakagawa-Sakata (MNS) matrix [16] is obtained from the Y e in eq. ( 36) and M ν in eq.( 40) as To obtain the observed neutrino masses [4], GeV, this condition is rewritten as which is satisfied with the natural GUT in Table I To obtain the MNS matrix as in eq. ( 41), which is consistent with the observations.Especially conditions 2,3, and 5 are critical for the difficulty to obtain larger h.This is the tension between the neutrino masses and the gauge coupling unification in the natural GUT scenario.The above conditions are considered to build explicit natural GUT models with suppression factors in the next section.Note that we have assumed that ψ 1 = ψ 3 + 3 and ψ 2 = ψ 3 + 2 to obtain realistic quark and lepton masses and mixings.

III. SOLUTIONS FOR TENSION BETWEEN NEUTRINO MASSES AND GAUGE COUPLING UNIFICATION
In this section, we examine several possibilities to avoid the tension between the neutrino masses and the gauge coupling unification.Since this tension is strongly dependent on the basic assumption that all terms allowed by the symmetry are introduced with O( 1) coefficients, we explore the possibilities in which some of the terms have much smaller coefficients than 1.When we introduce the terms with small coefficients, we require that the VEV relations for the GUT singlet operator O with the U(1) A charge o as do not change because they play critical roles in the natural GUT scenario.
After finding the set of terms with small coefficients that avoids this tension, we discuss the reason for the small coefficients, for example, because of an approximate symmetry etc.
Furthermore, we build concrete natural GUT models which can avoid the tension between the neutrino masses and the gauge coupling unification.And we discuss the nucleon decay in those models.
A. Suppression factor for terms for right-handed neutrino masses One of the easiest way to avoid the tension is to introduce small coefficients for the terms which give the right-handed neutrino masses as where we omit the O(1) coefficients.Since the right-handed neutrino masses become smaller, the (left-handed) neutrino masses become larger.The heaviest neutrino mass can be given as which must be the observed value m ντ ∼ 0.05 eV.Note that this suppression factor does not change the VEV relation in eq. ( 44), and the mass spectrum of superheavy particles except the right-handed neutrinos.This means that the beta functions do not change, and therefore, the gauge coupling unification conditions remain unchanged as h ∼ 0 and Λ = Λ G .
Since h = 0 allows the Higgs mass term H 2 which spoils the doublet-triplet splitting, we take h = −1.An concrete natural GUT model with h = −1 is given in Table II.Note that the half integer U(1) A charges for matter fields play the same role as the R-parity, and all requirements listed in the end of the previous section are satisfied in this model.

Concretly we introduce the following suppression factors
where the F -flatness conditions of the first four superpotentials determine the VEVs of negatively charged fields, while the last superpotential are important to fix the mass spectrum of superheavy particles.The mass matrices of 5 and 5 of SU( 5) become The determinants of reduced mass matrices, which are important to obtain the RGEs, are written as Similarly, the mass matrices of 10 of SU( 5) become and the determinants of reduced mass matrices are gives as For adjoint field G, W, X and X, mass matrices are given by and the determinants of the reduced mass matrices are When we define the suppression parameters as the gauge coupling unification conditions can be rewritten as (67) The heaviest neutrino mass can be written as eV. (69) In the next subsection, using the above results, we discuss several possibilities to avoid the tension between the neutrino masses and the gauge coupling unification.

C. Models
In this subsection, we build several explicit natural GUT models which have no tension between the neutrino masses and the gauge coupling unification by introducing various suppression factors as discussed in the previous subsection.
First, let us explain the features common to the natural GUT models built in this paper.
They have no tension between the neutrino masses and the gauge coupling unification while they have all advantages of the usual natural GUTs except the basic principle that all terms allowed by the symmetry are introduced with O(1) coefficients.We fix a = −1 and a ′ = 3 which allow terms A ′ A and A ′ A 3 , and forbid A ′ A 5 and more to obtain the DW type VEV naturally, although we have another options to take a = −1/2 and a = 3/2 which predict longer lifetime of nucleon via dimension 6 operators because of larger unification scale.To obtain the realistic natural GUTs, they must satisfy the conditions listed in the end of the previous section, which can be rewritten as And the three relations ( 67)-( 69) we obtained in the end of the last subsection are important to build the natural GUT models.
In this model, we assume that ε H ′ ≪ 1 and the others are O(1).The ε H ′ dependence of determinants of the reduced mass matrices become The eqs. ( 67),( 68) and ( 69) are rewritten as FIG. 1. proton decay mediated by colored Higgs Here, we try to build a natural GUT in which not only the tension is avoided but also the cutoff scale becomes larger than Λ G since the cutoff scale is quite important in predicting the nucleon lifetime.
In addition to ε the eqs.( 67),( 68) and (69) become Note that if λ δ = ε H ′ , the above conditions become nothing but those in the previous model.Here, for simplicity, we assume that the cutoff is around the reduced Planck scale as Λ ∼ M P lanck ∼ 2 × 10 18 GeV.As a result, the first two equations ( 83) and ( 84) become The condition (85) can be replaced as Generically, the effective colored Higgs mass can be obtained as where the last similarity is shown by eqs.( 67) and (68).Obviously, Therefore, to obtain the larger effective colored Higgs mass than Λ G , it is sufficient that we obtain the effective colored Higgs mass as The explicit U(1) A charge assignment can be shown in Table III, which is the same as in the model 2. From the eqs.( 67) and ( 68), we can obtain Since the unification scale λ −a Λ < Λ G , the nucloen decay via dimension 6 operators may be seen in future experiments, while the proton decay via dimension 5 operators is suppressed although it strongly depends on ε 2H ′ .
Unfortunately, these suppression factors cannot be realized by an approximate symmetry.
For example, an approximate symmetry, in which only H ′ has non-trivial charge, results in IV. DISCUSSION AND SUMMARY Under the natural assumption that all terms allowed by the symmetry are introduced by O(1) coefficients, the natural GUT solves various problem of SUSY GUT and gives a GUT that leads to the Standard Model, which is consistent with almost all observations and experiments.Unfortunately, the natural GUT has an unsatisfied point that many O(1) coefficients must be artificially chosen between 0.5 and 2 to achieve unification of the gauge coupling constants.This problem is due to the fact that the neutrino masses become too small to satisfy the measured values under the conditions for unification of the gauge coupling constants.
In this paper we discussed how to avoid the tension between the unification of gauge coupling constants and neutrino masses in the natural GUT.In particular, we considered the possibilities that the tension could be eliminated by assuming that, for some reason, some terms have suppression factors in addition to the suppression factors determined by the U(1) A symmetry.We found several solutions and explicitly built natural GUT models .
For some solutions, we also found that their additional suppression factors can be understood naturally with approximate symmetries.
We focused on how nucleon decay, an important prediction of GUT, changes in these solutions.In the original SUSY GUT scenario, the nucleon decay via dim.5 operators is important while the nucleon decay via dim.6 operators is suppressed because of the larger unification scale.In the original natural GUT scenario, the nucleon decay via dim.6 operators becomes interesting because the unification scale becomes generally lower, while the nucleon decay via dim.5 operators is strongly suppressed because the effective colored Higgs mass becomes λ 2h Λ G with h = −3.This is an important prediction in the natural GUT.In the natural GUT with suppression factors, which is discussed in this paper to avoid the tension between the gauge coupling unification and the neutrino masses, these predictions for nucleon decay can change.The model with suppression factor for terms for right-handed neutrino masses gives similar predictions on nucleon decay as the original natural GUT because the colored Higgs mass becomes λ 2h Λ G with h = −1.In the models with suppression factors explained by a symmetry for terms with positively charged fields, the nucleon decay via dim.5 operators becomes more important generically, while the nucleon decay via dim.6 operators can be suppressed.This is an important observation in this paper, although we also showed that the nucleon decay via dim.5 operators can be suppressed in a natural GUT with suppression factors which cannot be understood by an approximate symmetry.
Note that in the natural GUT, in which the suppression factors can be understood by an approximate symmetry, the suppression factor discussed above cannot be understood in terms of spontaneous breaking of the symmetry.For example, if we try to explain the suppression factor of the H ′ H term, where H ′ and H have odd and even Z 2 parity, respec- and β E c = 0. Thus the each 4 × 4 matrix has one vanishing eigenvalue.The mass spectrum of the remaining three modes is ( 2) R , U(1) B−L and U(1) Y , respectively.Since the model has the left-right symmetry above Λ C , we expect g 2 = g R at µ > Λ C .
into the SM gauge group G SM , an adjoint Higgs 45 A and one pair of spinor 16 C and anti-spinor 16 C are min- fields has no effects in fixing these VEVs of negatively charged fields under the assumption.Therefore, only the terms which include one positively charged field are important to fix the VEVs.The superpotential for fixings the VEVs are . Here we use h + 2ψ 3 = 0, which is required to obtain O(1) top Yukawa coupling.It is difficult to obtain larger h by smaller c and/or larger c, because several conditions are required to obtain realistic natural GUT 3 + h = 0: to obtain O(1) top Yukawa coupling, i.e., to obtain the term