It is shown how to use as horizontal symmetry the dicyclic group $Q_6 \subset
SU(2)$ in a supersymmetric unification $SU(5)\otimes SU(5)\otimes SU(2)$ where
one $SU(5)$ acts on the first and second families, in a horizontal doublet, and
the other acts on the third. This can lead to acceptable quark masses and
mixings, with an economic choice of matter supermultiplets, and charged lepton
masses can be accommodated.Comment: 10 pages, LaTe

C. Froggatt

D.B. Kaplan

D.S. Shaw

E. Baver

E. Dudas

E. Papageorgiu

H. Georgi

H. Murayama

J. Ellis

J.S. Hagelin

L. Ibanez

L.J. Hall

M. Dine

M. Leurer

M. Schmaltz

O.C.W. Kong

Otto C. W. Kong

P. Binétruy

P. Pouliot

P. Ramond

P.H. Frampton

Paul H. Frampton

R. Barbieri

S. Coleman

S. Hawking

S.B. Giddings

T. Banks

V. Ben-Hamo

V. Jain

Y. Nir

English

arXiv

It is shown how to use as horizontal symmetry the dicyclic group $Q_6 \subset SU(2)$ in a supersymmetric unification $SU(5)\otimes SU(5)\otimes SU(2)$ where one $SU(5)$ acts on the first and second families, in a horizontal doublet, and the other acts on the third. This can lead to acceptable quark masses and mixings, with an economic choice of matter supermultiplets, and charged lepton masses can be accommodated. Comment: 10 pages, LaTe

Frampton, Paul H.

Kong, Otto C.W.

Carolina Digital Repository

arXiv:hep-ph/9511343v1  16 Nov 1995October 1995IFP-715-UNCDicyclic Horizontal Symmetry and Supersymmetric GrandUnification.Paul H. Frampton and Otto C. W. KongInstitute of Field Physics, Department of Physics and Astronomy,University of North Carolina, Chapel Hill, NC 27599-3255AbstractIt is shown how to use as horizontal symmetry the dicyclic group Q6 ⊂ SU(2)in a supersymmetric unification SU(5)⊗SU(5)⊗SU(2) where one SU(5) actson the first and second families, in a horizontal doublet, and the other actson the third. This can lead to acceptable quark masses and mixings, with aneconomic choice of matter supermultiplets, and charged lepton masses can beaccommodated.Typeset using REVTEX1The smallness of most of the quark masses and mixing parameters and the strong hier-archy among them is one of the most interesting puzzles in particle physics. Spontaneouslybroken horizontal symmetry is the most popular candidate theory for understanding theflavor structure, including in supersymmetric models. In the context of the MSSM, a hori-zontal symmetry may also give a viable alternative to build in a super-GIM mechanism tosuppress FCNC induced by supersymmetric particles [1–3]. Attempts has also been made touse horizontal symmetry to address the µ-problem [2,4], the strong CP problem [5], FCNCdue to light leptoquarks [6], and baryon number violation in supersymmetry [7]. There ishence a growing interest in the topic.However, as global symmetries are in general not respected by gravitational effects [8],the horizontal symmetry should be gauged. Canceling the gauge anomalies then imposes astrong constraint on model building [9–12]. For a simple nonabelian symmetry, we are leftwith essentially only SU(2) and its discrete dicyclic subgroups Q2N [12–15].Now we consider an extra desirable ingredient, compatibility with supersymmetric verti-cal (grand) unification, like SU(5). The only GUT-compatible gauged horizontal symmetrymodel proposed so far is incompatible with SUSY [12]. Here we provide the first SUSY-GUTcompatible such model.Inspired by the anti-unification approach to quark masses [16], models with separate GUTgroups for each of the three families has been introduced [17]. Here we consider instead onlytwo SU(5)’s for horizontal singlet and doublet families. The structure then gives, to thefirst approximation, rank one quark mass matrices. We show that, with judiciously chosenheavy scalar VEVs, the full hierarchical and phenomenologically-viable quark mass matrixtextures can be generated, using nonrenormalizable gravitational interactions [18].Our model has gauged SU(5)⊗SU(5)⊗SU(2), with this symmetry broken to a diagonalSU(5) (SUSY-)GUT group around and above the GUT scale. The full pattern of symmetrybreaking is illustrated in Figure 1.The assignment of the three families of quarks and leptons to (SU(5)⊗ SU(5)⊗Q6) isthus23rd family (5̄ + 10, 1, 1)1st and 2nd families (1, 5̄ + 10, 21).Upon breaking to diagonal SU(5) this becomes a normal 3-family SUSY-GUT.The Higgses which will break electroweak symmetry are in (5+5̄, 1, 1) and so couple onlyto the third family in a renormalizable fashion. Scalar VEVs in (5̄, 5) or (5, 5̄) will breakto the diagonal subgroup. There will also be SU(5)⊗ SU(5) singlets, non-trivial under Q6.Beyond these scalars, it will be necessary only to introduce an extra (15, 1̄0, 21) multiplet tocomplete the model.Taking as an expansion parameter λ ∼ sinθc ∼ .22 we will use two scale below MP lanckwhich are taken as M1 ∼ λM̃P lanck which characterizes the VEV of a (1, 1, 21) and M2 ∼λ3M̃P lanck which sets the SU(5) ⊗ SU(5) breaking VEVs. In fact, M2 lies just above theusual MGUT ≈ 2×1016GeV , as the effective Planck mass M̃P lanck is given by MP lanck/√8π ≈2.4×1018GeV . Thus, the hierarchy of the observed quark masses at accessible energy merelyreflect the existence of the superheavy scalars lying at and above MGUT .To keep track of the book-keeping for the components of the Q6 couplings, we find itmost convenient to assign to the two components of a 2n doublet the values ±n, reflectingthe eigenvalues of 2T3 = ±n in the natural embedding SU(2) ⊃ Q2N . Recall that foreven-dimensional SU(2) irreducible representationsd −→ 21 + 23 + .... 2d−1while for odd d,d −→ singlet + 22 + 24 + .... 2d−1;the singlet is 1 for d = 1, 5, 9, .... and 1′for d = 3, 7, 11, .... . Of course we will need onlyd = 1, 2, 3 and 4 for Q6.With this book-keeping, we find that the mass matrix textures emerge from VEVs asfollows:3• the only scalar with VEV at scale M1 is (1, 1, [+1])• the VEVs at scale M2 are– (5̄, 5, [−3])– (5, 5̄, [−2]) and (5, 5̄, [−1])– (15, 1̄0, [−1])where the Q6 entry implies the 2T3 eigenvalue. Tracking down all the entries of the massmatrices to the lowest order in λ, we have the follwing result:Mu ∼λ8 λ6 λ9λ6 λ4 λ7λ9 λ7 1(1)Md ∼λ6 λ4 λ5λ4 λ3 λ3λ7 λ5 1(2)The authors of [20] have analyzed all possible symmetric quark mass matrices with themaximal (six) and next-to-maximal (five) number of texture zeros, and concluded that onlyfive models, denoted by the roman numerals I to V in their work, are phenomenologicallyviable. Note that the symmetric structure is just an input assumption. In our case, theGUT structure enforced a symmetric mass matrix for the up-sector, but leaves that for thedown-sector arbitrary. While U(1) flavor symmetry constructions for quark mass matriceswith nonsymmetric hierarchical textures have been attempted [2,10,19], the full list of suchphenomenologically viable quark mass matrices is not yet available. However, we can simplyexploit the fact that low-energy physics is unaffected by an arbitrary rotation of the right-handed quark fields. Discarding the large order entries and imposing a rotation on theright-handed second and third down-quark fields with an angle ∼ λ3, we obtain, in thesymmetric basis,4Mu ∼0 λ6 0λ6 λ4 00 0 1(3)Md ∼0 λ4 0λ4 λ3 λ30 λ3 1(4)corresponding to case I of ref. [20], hence showing that the (asymmetric) quark mass textureis phenomenologically viable.Now we turn to the charged lepton mass matrix. The simplest way to accommodate itto obtain the Georgi-Jarlskog pattern [21] by replacing the scalar VEV (5, 5̄, [−2]), whichis responsible for the (Md)22 entry, with a (5, 4̄5, [−2]). If one wants to avoid having a 4̄5,there is the alternative suggested by Ellis and Gaillard [18]. While the diagonal SU(5)singlet from the (5, 5̄) contributions to the quark and lepton masses are the same, the otherSU(3) ⊗ SU(2) ⊗ U(1) singlet in the adjoint 24 of the diagonal SU(5) gives quark andlepton masses in the ratio −3/2. Ellis and Gaillard showed that if both the singlet and the24 contribute, with partial cancellation in the lepton-sector, the 5̄ VEV could fit both thequark- and lepton-sectors. The 24 VEV is of course GUT-breaking, which is needed anyway.In our case, its contribution has to be smaller by about a factor of MGUT/M2. Without goinginto detail, a simple comparison with the Ellis-Gaillard analysis shows that this is can besuccessful.Having discussed both the quark and charged-lepton mass matrix texture construction,a few comments are in order:• The breaking of the horizontal SU(2) through the discrete dicyclic subgroup Q6 isneeded to avoid the otherwise large D-term contributions to the scalar quark massesin the SU(2) breaking [3]. Our model may otherwise be considered only in the SU(2)framework. However, the D-term contributions would lift any assumed degeneracy5among the squarks and cause unacceptable FCNC in for example K − K̄ mixing (see[22] and references therein). The strongest FCNC constraint can be expressed as anupper limit on the (12) entry of the left-handed down-squark m̃2LLmatrix, in thequark-mass eigenbasisδm̃2ds= m̃21K11K†12 + m̃22K12K†22 + m̃23K13K†32 (5)where m̃2iare the three eigenvalues and K the unitary transformation matrix thatdiagonalizes m̃2LL. In the limit that K13K†32 is negligible, this reduces toδm̃2ds ≈ (m̃22 − m̃21)K12 (6)hence a degeneracy condition between m̃21and m̃22, unless the mixing K12 is itself ex-ceedingly small [23]. As noted in ref. [3], the 2+1 family structure, gives a natural firstorder degeneracy between m̃21and m̃22, and is therefore flavorable from the perspective.The degeneracy is however lifted as the horizontal symmetry is broken. In our model,the lifting is of order λ2 which is too large. Ex tra mechanisms, as proposed in ref. [3],is needed to help suppress the FCNC.• In principle, the non-renormalizable mass terms may be obtained, alternatively, fromthe Froggatt-Nielsen mechanism [24]. In that case, one needs M1/M0 ∼ λ andM2/M0 ∼ λ3 where M0 is the mass scale of the vector-like fermions mediating theYukawa vertices involving the chiral fermions. However, M0 cannot really be broughtdown much below MP lanck because the proliferation of heavy supermultiplets maylead to a non-perturbative gauge coupling [2].• The supermultiplets that contain the SU(5) ⊗ SU(5) breaking VEVs in the modelare assumed to be vector-like. Hence they are heavy and have no contribution togauge anomalies. The supermultiplet (1, 1, 21) can have heavy Majorana mass. It hasa contribution which helps to cancel the otherwise non-trivial global-SU(2) anomaly.Local gauge anomaly cancellation in our model is completely straightforward with noadditional states.6• The (1, 1, 21) can be identified naturally as a right-handed neutrino supermultiplet.If an extra (1, 1, 1) is added, the family structure of the right-handed neutrinos isthen the same as the quarks and leptons. While this appears natural, the neutrinomasses and mixings hence derived, assuming no extra VEVs, do not look very good.However, the right-handed-neutrino-sector need not have the same family structure asthe quarks and leptons, so long as the global-SU(2) anomaly condition is satisfied; andthere could be some extra multiplets with or without VEVs among them that modifythe neutrino masses and mixings without upsetting the quark and chagred-lepton masstextures.• Like the minimal SUSY-SU(5), the model has dimension-4 baryon number violatingoperators that have to be removed by imposing R-parity or otherwise. In particular the(5̄, 1, 1) Higgs supermultiplet definitely has to be distinguished from the quark-leptonone in the same representations to avoid fast proton decay. Finally, the infamousdoublet-triplet splitting problem has not been addressed.This work was supported in part by the U.S. Department of Energy under Grant DE-FG05-85ER-40219, Task B.Figure Caption.Fig.1 Illustration of the symmetry breaking pattern of the model.7REFERENCES[1] M. Dine, R. Leigh and A. Kagan, Phys. Rev. D, 48, 4269 (1993).[2] Y. Nir and N. Seiberg, Phys. Lett. B, 309, 337 (1993); M. Leurer, Y. Nir and N. Seiberg,Nucl. Phys. B420, 468 (1994).[3] A. Pomoral and D. Tommasini, CERN-TH/95-207.[4] V. Jain and R. Shrock, Stony Brook preprint ITP-SB-95-22, hep-ph/9507238; Y. Nir,Phys. Lett. B, 354, 107 (1995).[5] T. Banks, Y. Nir and N. Seiberg, preprint RU-94-14, WIS-94-14-PH, hep-ph/9403203.[6] E. Baver and M. Leurer, Phys. Rev. D, 51, 260 (1995).[7] H. Murayama and D.B. Kaplan, Phys. Lett. B 336, 221 (1994); V. Ben-Hamo and Y.Nir, Phys. Lett. B 339, 77 (1994).[8] S. Hawking, Phys. Lett. B 195, 377 (1987); Phys. Rev. D 37, 904 (1988); S.B. Giddingsand A. Strominger, Nucl. Phys. B306, 890 (1988); ibid. B307, 854 (1988); S. Coleman,Nucl. Phys. B307, 867 (1988); ibid. B310, 643 (1988).[9] L. Ibanez and G. G. Ross, Phys. Lett. B 332, 100 (1994); E. Papageorgiu, Z. Phys.C64, 509 (1994); P. Binétruy and P. Ramond, Phys. Lett. B 350, 49 (1995); V. Jainand R. Shrock, Phys. Lett. B 352, 83 (1995).[10] E. Dudas, S. Pokorski and C.A. Savoy, Phys. Lett. B, 356, 45 (1995).[11] D.S. Shaw and R.R. Volkas, Phys. Rev. D 47, 241 (1993);[12] P.H. Frampton and O.C.W. Kong, Phys. Rev. Lett. 75, 781 (1995).[13] P. H. Frampton and T. W. Kephart, Phys. Rev. D 51, R1 (1995); Int. J. Mod. Phys. A(1995, in press)[14] For global version of nonabelian horizontal symmetry, see P. Pouliot and N. Seiberg,8Phys. Lett. B, 318, 169 (1993) D.B. Kaplan and M. Schmaltz, Phys. Rev. D, 49, 3741(1994); M. Schmaltz, Phys. Rev. D, 52, 1643 (1995).[15] For a semi-simple flavor group, see L.J. Hall and H. Murayama, Berkerley preprintUCB-PTH-95-29, hep-ph/9508296.[16] C. Froggatt, G. Lowe and H. Nielsen, Phys. Lett. B 311, 163 (1993); Nucl. Phys. B414,579 (1994); ibid. B420, 3 (1994).[17] R. Barbieri, G. Dvali and A. Strumia, Nucl.Phys. B435 102 (1995).[18] J. Ellis and M.K. Gaillard, Phys. Lett. B, 88, 315 (1979).[19] M. Leurer, Y. Nir and N. Seiberg, Nucl. Phys.B398, 319 (1993).[20] P. Ramond, R.G. Roberts and G.G. Ross, Nucl. Phys. B406, 19 (1993).[21] H. Georgi and C. Jarlskog, Phys. Lett. 86B, 297 (1979).[22] J.S. Hagelin, S. Kelley and T. Tanaka, Nucl. Phys. B415, 293 (1994).[23] This is the quark-squark alignment scenario suggested in ref. [2].[24] C. Froggatt and H. Nielsen, Nucl. Phys. B147, 277 (1979).9FIGURESSU(5)⊗ SU(5)⊗ SU(2)SU(5)⊗ SU(5)⊗Q6SU(5)⊗ SU(5)SU(5)DSU(3)c ⊗ SU(2)L ⊗ U(1)YM1 ∼ λM̃P lanckM2 ∼ λ3M̃P lanckMGUTMSUSYMW❄❄❄❄❄FIG. 1. Illustration of the symmetry breaking pattern of the model10

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