The minimal supersymmetric grand unified theory

Physics Letters B 588 (2004) 196–202 www.elsevier.com/locate/physletb

The minimal supersymmetric grand unified theory C.S. Aulakh a , B. Bajc b , A. Melfo c , G. Senjanovi´c d , F. Vissani e a Department of Physics, Panjab University, Chandigarh, India b J. Stefan Institute, 1001 Ljubljana, Slovenia c Centro de Astrofísica Teórica, Universidad de Los Andes, Mérida, Venezuela d International Center for Theoretical Physics, Trieste, Italy e INFN, Laboratori Nazionali del Gran Sasso, Theory Group, Italy

Received 28 November 2003; received in revised form 19 February 2004; accepted 11 March 2004 Available online 9 April 2004 Editor: G.F. Giudice

Abstract We show that the minimal renormalizable supersymmetric SO(10) GUT with the usual three generations of spinors has a Higgs sector consisting only of a “light” 10 dimensional and “heavy” 126, 126 and 210 supermultiplets. The theory has only two sets of Yukawa couplings with fifteen real parameters and ten real parameters in the Higgs superpotential. It accounts correctly for all the fermion masses and mixings, neutrinos included. The theory predicts at low energies the MSSM with exact R-parity. It is arguably the minimal consistent supersymmetric grand unified theory.  2004 Published by Elsevier B.V. PACS: 12.10.Dm; 12.10.Kt; 12.60.Jv

1. Introduction After more than twenty years of low energy supersymmetry and grand unification the minimal supersymmetric grand unified theory (GUT) is still commonly assumed to be based on the SU(5) gauge group with three generations of 5¯ and 10 dimensional matter supermultiplets, and 5H , 5¯ H and 24H dimensional Higgs supermultiplets [1]. The “Higgs” superpotential contains 4 complex parameters. It has two sets of Yukawa couplings, and as is well-known predicts equal down quark and charged lepton masses at the GUT scale, relations which only work for the third

E-mail address: [email protected] (F. Vissani). 0370-2693/$ – see front matter  2004 Published by Elsevier B.V. doi:10.1016/j.physletb.2004.03.031

generation and fail badly for the first two. This can be corrected, e.g., by 1/MPl terms, but at least two new sets of Yukawa couplings should be included. On top of that, this minimal GUT is plagued by R-parity violating couplings. If these are set to zero, neutrinos are predicted to be massless, which in view of atmospheric and solar neutrino data seems an untenable position. To correct for that, there are the following options: (i) add the effective, nonrenormalizable, couplings fij 5¯ i 5¯ j 5H 5H /M, where M
MGUT . This fails since the data strongly suggest M < MGUT ; (ii) add the right-handed neutrinos, which means their 3 masses and 9 complex Dirac–Yukawa couplings;

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(iii) add 15H and 15H dimensional Higgs superfields with 6 complex Yukawa couplings, namely fij 5¯ i 5¯ j 15H , and 4 more complex couplings among 15H , 15H , 5H , 5¯ H and 24H . Let us count the number of parameters for any of these possible versions of the minimal realistic supersymmetric SU(5) theory. We can diagonalize the down quark (charged lepton) Yukawa matrix, which leaves 3 real parameters. The symmetric up-quark Yukawas with the freedom of a global U(1) rotation of 5H give us 6 × 2 − 1 = 11 real parameters. With the 5¯ H and 24 fields in WH we can redefine two phases, thus 4 × 2 − 2 = 6 real parameters in WH . So, with massless neutrinos we have already 14(WY ) + 6(WH ) + 1(gauge) = 21 real parameters. Add neutrino masses and you get: (i) 6 ×2 = 12 more real parameters in WY → 33 real parameters in all; (ii) 3 + 9 × 2 = 21 more real parameters in WY → 42 real parameters in all; (iii) 6 × 2 = 12 more real parameters in WY , 4 × 2 − 2 = 6 real parameters in WH and then the total is 39 real parameters. As we explained above (i) should be discarded; a realistic theory has at least 39 parameters. We keep (i), though, in order to emphasize the minimality of the SO(10) theory which we discuss below. It is instructive to compare such a SU(5) GUT with the minimal supersymmetric standard model (MSSM) with massive neutrinos. By this we mean effective neutrino mass operators of the type (i) mentioned in the case of SU(5), which amounts to effective Majorana masses for neutrinos. As usual, we assume R-parity (otherwise you have many more parameters). We have 6 quark masses, 3 quark mixings and 1 CP phase, 6 lepton masses, 3 lepton mixings and 3 CP phases, so in total 22 real parameters in WY . With 3 gauge couplings and 1 real parameter (µ term) in WH , we get in total 26 real parameters in MSSM. Notice that we did not count the soft terms, they ought to be included separately, but they are present in any theory.

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In this work we advocate as the minimal supersymmetric grand unified theory the renormalizable SO(10) theory based on the 210 dimensional representation originally introduced in [2,3] shortly after the first proposal based upon the 45 and 54 representations [1]. We find that this theory has 26 real parameters, as in the MSSM, and much less than in SU(5). And yet it consistently describes all the low-energy phenomena, in particular the fermionic masses and mixing angles. It is also a theory of R-parity whose conservation it predicts to all orders in perturbation theory. In short, it is the minimal supersymmetric grand unified theory based on a single gauge group without additional ad hoc symmetries. The further tests of the theory will be provided by d = 5 proton decay, leptogenesis and flavour violation processes. At first glance, this result of SO(10) being a minimal theory, more economical, simpler than SU(5), may appear as a contradiction in terms. After all, SU(5) is a smaller gauge group. Here, it may be useful to recall a lesson from the SO(3) model of leptons of Georgi and Glashow [4]. In spite of being a smaller gauge group than the SU(2) × U(1) theory, it is a much more complicated theory with many more parameters in the Yukawa sector. In this sense, the SM is the minimal electro-weak theory, not just the minimal correct one.

2. The theory We will consider a renormalizable SO(10) theory. Why not take 1/MPl terms? After all they could play a nontrivial role in fermion masses and mixings. The point is that in this case we lose both minimality and predictivity. Take for example the supersymmetric SO(10) theory based on 16H , 16H , 45H , 10 and nonrenormalizable superpotentials [5]. Even at order 1/MPl this theory has far more parameters unless it is augmented with extra discrete (or continuous) symmetries. One such discrete symmetry is needed anyway in order not to break R-parity at MGUT . It is well known that the renormalizable theory ¯ supermultiplets: 126 requires 126 (Σ) and 126 (Σ) alone produces right-handed neutrino masses and does not lead to anomalies, but its vacuum expectation value (VEV) leads to a nonvanishing D-term, and

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therefore to supersymmetry breaking; adding 126, supersymmetry can be preserved at a high scale. These VEVs are SU(5) singlets, so they leave SU(5) unbroken: thus, more Higgs fields are needed. The minimal choice is the four-index antisymmetric representation 210 (Φ), that was first considered in [2,3]. Notice that neither 45 nor 54 by themselves can do the job at the renormalizable level; both are needed [6], leading to more parameters in the superpotential. The large representations required to account for fermion mass matrices in renormalizable theories predict a Landau pole in the gauge coupling at a scale ΛU an order of magnitude or so above MGUT . This indicates a new strongly coupled dynamics not far from the unification scale which may be analyzed taking advantage of the exact supersymmetry there [7]. The scale ΛU as much as MPl could easily leave imprints at the GUT scale. We decided to ignore them in order to be concrete. The theory so defined makes clear predictions, and experiment will tell us how good they are. The superpotential involving the Φ, Σ and Σ¯ fields [2,3,8] mΦ λ Φij kl Φij kl + Φij kl Φklmn Φmnij WH = 4! 4! η mΣ ¯ Σij klm Σij klm + Φij kl Σij mno Σ¯ klmno + 5! 4! (1) has only 4 independent parameters. Although the representations are large, the number of parameters is small; the theory is simple and economical. We argue in passing against the usual phobia of large representations: it is not size that matters! It is convenient, especially when discussing fermion masses as we do below, to decompose the Higgs superfields under the Pati–Salam maximal subgroup SU(2)L × SU(2)R × SU(4)C (for a useful connection between SO(10) and Pati–Salam see [9]): Φ ≡ 210 = (1, 1, 15) + (1, 1, 1) + (1, 3, 15) + (3, 1, 15) + (2, 2, 6) + (2, 2, 10) + (2, 2, 10), Σ ≡ 126 = (1, 3, 10) + (3, 1, 10) + (1, 1, 6) + (2, 2, 15), ¯ Σ ≡ 126 = (1, 3, 10) + (3, 1, 10) + (1, 1, 6) + (2, 2, 15).

The physically allowed VEVs we call p for (1, 1, 1), a for (1, 1, 15), ω for (1, 3, 15) and σ and σ¯ for (1, 3, 10) and (1, 3, 10). After some straightforward computation, the superpotential as a function of these VEVs becomes
 WH = mΦ p2 + 3a 2 + 6ω2
 + 2λ a 3 + 3pω2 + 6aω2 + mΣ σ σ¯ + ησ σ¯ (p + 3a − 6ω).

(2)

The results agree with those of [8]. Vanishing of the D-terms (⇒ |σ | = |σ¯ |) and the F-terms determines the supersymmetric vacua. The SU(5) symmetric vacuum p = a = −ω and the L–R symmetric one p = ω = σ = 0, a = −mΦ /λ have simple expressions, while the standard model vacuum is p=−

mΦ x(1 − 5x 2 ) , λ (1 − x)2

ω=−

mΦ x, λ

σ σ¯ =

a=−

mΦ 1 − 2x − x 2 , λ 1−x

2m2Φ x(1 − 3x)(1 + x 2 ) , ηλ (1 − x)2 (3)

where x is a solution of the cubic equation: 8x 3 − 15x 2 + 14x − 3 = −(x − 1)2

λmΣ . ηmΦ

(4)

The most typical case has single step breaking. E.g., when λmΣ ∼ ηmΦ (x ∼ 0.21), we get p ∼ −0.67 mΦ /λ, ω ∼ −0.21 mΦ /λ, −0.27 mΦ /λ, a ∼ √ σ ∼ σ¯ ∼ 0.51 mΦ / ηλ, that corresponds to an approximate single step breaking of SO(10) → MSSM. However, in some cases σ , and thus the scale of right-handed neutrino masses, is smaller than the other VEVs. E.g., this happens when x ∼ 1/3, that is 3λmΣ ∼ −2ηmΦ . Since the vacua are discrete (no flat directions) one expects no additional massless states beyond the MSSM ones, i.e., no pseudo Goldstone bosons. It is always a good exercise to check this. We have performed it and the result confirms the physical reasoning, apart from special values for x. These special values are of no physical importance: for example, they are not needed for the doublet–triplet splitting or d = 5 proton decay suppression. Recently a detailed analysis of this has appeared, where this is made explicit [10,11]. In particular, it can be seen that

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after the usual fine-tuning one ends up with only one pair of Higgs doublet superfields (see Section 3). As usual a 10 dimensional (H ) Higgs multiplet is needed for fermion masses, and the superpotential gets an additional piece WH = mH H 2 +

1 Φij kl Hm (αΣij klm + α¯ Σ¯ ij klm ) 4! (5)

with three more complex parameters, seven in total. With the matter fields being 16 dimensional spinors Ψa (a = 1, 2, 3 is a generation index), we have two symmetric Yukawa couplings in generation space ¯ WY = Ψ (Y10 H + Y126 Σ)Ψ.

(6)

We diagonalize one of them, which amounts to 3 real couplings; the other one then has 6 × 2 = 12 more real parameters. With four Higgs superfields we can redefine four phases of the seven complex couplings and so we have 7 × 2 − 4 = 10 real parameters in the Higgs superpotential WH . Together with the Yukawa and gauge sector, this means 15 + 10 + 1, that is 26 real parameters in minimal SO(10), just as in the MSSM. This is a remarkable result. Furthermore, the theory is in perfect accord with all the quark and lepton masses and mixings, as we now discuss.

3. Fermion masses and mixings Naively, one imagines only the field (2, 2, 1) in H to have a VEV at the electroweak scale. The situation is more subtle. There are 4 Higgs doublet pairs (with Y = ±1) which mix: one pair each from (2, 2, 1) in the 10, (2, 2, 15)s in the 126 and 126 and (2, 2, 10 + 10) in the 210. It is easy to see that (2, 2, 15) fields in 126 mix with (2, 2, 1) [9] through the VEVs of 210, which are of order MGUT . The usual minimal fine tuning is achieved simply by choosing mH = mΦ

p10 α α¯ 2ηλ (x − 1)p3 p5

(7)

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where the three polynomials of x are given by [11] p3 = 12x 3 − 17x 2 + 10x − 1,

(8)

p5 = 9x + 20x − 32x + 21x − 7x + 1, 5

p10 = 90x

4

10

3

2

− 858x + 2009x − 3073x 9

8

(9)

7

+ 4479x 6 − 5018x 5 + 3618x 4 − 1545x 3 + 377x 2 − 50x + 3.

(10)

This guarantees that only two Higgs doublets are light and it is easy to see that they are combinations of all doublets with the same hypercharge (for details see [11]). The crucial outcome is that all the color singlet bidoublets and in particular both (2, 2, 1) and (2, 2, 15) get a VEV and thus contribute to fermion masses. The idea to use 210 for this purpose was suggested a long time ago [3,12]. The predictions for the fermion masses were studied extensively [12–14], and the outcome is that it is possible to reproduce the masses and mixing angles of the charged fermions. What about neutrino masses? It is surprising that the theory is consistent with the data, even when one specifies the nature of the see-saw mechanism. The see-saw may proceed through right-handed neutrino masses (type I see-saw) [15] or through the VEV of the (3, 1, 10), which necessarily gets induced (type II see-saw) [16]. The two limiting cases mean one parameter less and yet both work and both predict θ13 to be quite large [17]. The type II case is particularly interesting, since the large atmospheric mixing angle is intimately tied to b–τ unification [18]. The VEVs of the (3, 1, 10) and (3, 1, 10) fields which generate the type II neutrino masses arise due to the coupling between 210, 126 and 126 (the coupling η in Eq. (1)). This mechanism was initially studied in [19] in the context of the Pati–Salam theory, where it was shown that one needs (2, 2, 10) and (2, 2, 10) multiplets for the purpose. This is another important role played by the 210 representation, since it automatically contains these fields. Among all the terms in the superpotential (1) one has for the relevant ones WH = η(3, 1, 10)(2, 2, 10)(2, 2, 15) + ML (3, 1, 10)(3, 1, 10) + · · · .

(11)

From F(3,1,10) = 0 one finds in a symbolic notation 

 (2, 2, 10)(2, 2, 15) (3, 1, 10) = η . ML

(12)

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Since (3, 1, 10) lives in 126, which has Yukawa couplings with fermions, through its VEV, it gives directly a mass to left-handed neutrinos. This is precisely type II see-saw. The relative importance of the type II see-saw is inversely measured by ML =

ηm x(4x 2 − 3x + 1) λ (1 − x)2

(13)

which is small for √ 3±i 7 x → 0 or x → 8

a light pseudo-majoron (with mass inversely proportional to the right-handed neutrino mass) that is incompatible with the measured Z decay width [22]: Thus, the soft terms must be such that the symmetry breaking preserves R-parity. Note that a similar a posteriori argument is invoked to avoid charge and colour breaking minima in the MSSM. Furthermore, the result we demostrated is completely analogous to the impossibility of breaking R-parity spontaneously in the MSSM [23] (where the majoron is strictly massless).

(14)

and in both cases type II dominates. In the language of the maximal subgroup SU(5), we say that the 126 multiplet of SO(10) Higgs is useful for three reasons: it provides us with the singlet 1 that we need for right-handed neutrinos, but also with 15 and 45 multiplets that are needed to describe the masses of the observed fermions via the left triplet VEV and the Georgi–Jarlskog mechanism [20], respectively. Also 210 has a threefold role: it permits the breaking of the SO(10) symmetry, and it allows 45 and 15 to couple with the usual 5 and 5¯ multiplets, thus getting a VEV.

4. The fate of R-parity We now discuss the situation regarding R-parity at the GUT scale (see also [3,21]). Since it is equivalent to matter parity, M ≡ (−1)3(B−L) , we discuss M in what follows. Under M, 16 is odd and the rest even. On the other hand, SO(10) has a Z4 center: 16 → i16, 10 → −10, 210 → 210, 126 → −126, 126 → −126. Clearly, M ∈ Z4 and thus R-parity is an automatic consequence of gauge SO(10). Since Φ, Σ and Σ¯ are even under M, R-parity continues to be an exact symmetry so far. The theory is even more predictive. It uniquely determines the low-energy effective theory: it is MSSM with exact R-parity. As we have seen, R-parity remains unbroken at the first stage of symmetry breaking. The question is whether this remains true at the electroweak scale, since in principle light sneutrinos could get a VEV, breaking R-parity eventually. In absence of a theory of the soft terms, one could doubt that this possibility can be ever excluded. But such a spontaneous breakdown of R-parity would produce

5. Proton decay Another important aspect of the theory regards proton decay. For simplicity, here we assume a single step breaking from the GUT symmetry down to the MSSM. The GUT scale, neglecting the GUT threshold effects, is then close to 1016 GeV [24]. In this case d = 6 induced proton decay is on the slow side (for a recent attempt to increase this rate see for example [25]). On the other hand the d = 5 decay tends to be quite fast [26]. For example in minimal SU(5) the situation is dramatic [27] although there are ways out [28]. This will clearly be one of the central tests of the theory. The d = 5 nucleon decay rate depends upon the mass matrix [9] of the five colour triplet antitriplet pairs coming from the 10, 126, 126, 210 irreps [29]. A detailed discussion of the decay rate is beyond the scope of this note, but it is of course of extreme importance, and is being performed in detail. A related and maybe equally important issue is the unification of gauge couplings. Due to the large number of particles a high precision analysis becomes very involved and is certainly beyond the scope of this Letter. Only recently, months after the first version of this work was issued, the particle spectrum was computed [11]. Armed with this one can perform a detailed study of all the threshold effects [30].

6. Leptogenesis An important appealing feature of the see-saw mechanism in general and SO(10) in particular is the celebrated mechanism of leptogenesis [31]. Again, the situation is quite complicated. Most works assume that the type I see-saw mechanism is responsible both

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for neutrino masses and leptogenesis. However as discussed above the type II see-saw seems favoured in this theory. A recent analysis [32] has shown that the contribution of the left-handed triplet cannot be neglected in such a case even if it is much heavier than the lightest right-handed neutrino N1 . Under the assumption that right-handed neutrinos have a hierarchical pattern, only the decay of N1 is important. Without the triplet the asymmetry can be written as  3 MN1 il Im[(YN )1i (YN )1l (MνI∗ )il ]  , (15) εN1 = 2 16π v 2 i |(YN )1i | where (MνI∗ )il is the type I see-saw mass matrix. On the other hand, under the assumption that the triplet is sizably heavier than N1 , its contribution to the asymmetry can be written [32] as  1 MN1 il Im[(YN )1i (YN )1l (MνII∗ )il ]   εN1 = − , 2 8π v 2 i |(YN )1i | (16) where (MνII∗ )il is the type II seesaw mass matrix. The above lepton asymmetries differ just by their respective contribution to the neutrino mass matrix (up to a factor −2/3). As a result, if the triplet contribution dominates the neutrino mass matrix, it is likely to dominate also the lepton asymmetry production. Since the minimal SO(10) theory constrains the otherwise arbitrary Yukawa couplings in the above formulae, a detailed analysis is called for.

Acknowledgements The work of C.A., A.M., G.S. and B.B. is supported by Department of Science and Technology of the Government of India (DST grant SP/S2/K-07/99), CDCHT-ULA (Project C-1073-01-05-A), EEC (TMR contracts ERBFMRX-CT960090 and HPRN-CT-200000152) and the Ministry of Education, Science and Sport of the Republic of Slovenia, respectively. A.M. and F.V. thank ICTP and INFN exchange program (F.V.), for hospitality during the course of this work. We thank Z. Berezhiani, G. Dvali and R.N. Mohapatra for useful discussions, A. Girdhar for collaboration in the first stages of this work, and D. Wyler for a careful reading of the manuscript and comments. We are

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grateful to K.S. Babu for pointing us to Ref. [8] after the first version of this Letter was released.

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