We discuss proton decay induced by dimension--5 operators in supersymmetric
models containing extra hypercharge-1/3 color-triplets. We derive a general
formula relating dimension-5 operator to the color-triplet mass matrix. We show
that certain zeros in the triplet mass-matrix together with some triplet
coupling selection rules can lead to elimination of dimension-5 operators. We
apply this mechanism to SU(5) and flipped SU(5) theories with extended Higgs
sectors.Comment: 6 pages, uses JHEP.cls, talk given at the Trieste Meeting of the TMR
  Network on Physics Beyond the Standard Model, 24-27 February 1999, Trieste,
  Italy (to appear in the proceedings

Gomez, Mario E.

English

arXiv

M. Gomez

Crossref

Yukawa couplings and proton decay in SUSY models

We discuss proton decay induced by dimension--5 operators in supersymmetric models containing extra hypercharge-1/3 color-triplets. We derive a general formula relating dimension-5 operator to the color-triplet mass matrix. We show that certain zeros in the triplet mass-matrix together with some triplet coupling selection rules can lead to elimination of dimension-5 operators. We apply this mechanism to SU(5) and flipped SU(5) theories with extended Higgs sectors

Gómez, M E

CERN Document Server

European Network on Physics beyond the Standard Model, 1999PROCEEDINGSYukawa couplings and proton decay in SUSY modelsMario E. GomezPhysics Division, School of Technology,Aristotle University of Thessaloniki,Thessaloniki, GR-540 06, GreeceE-mail: mgomez@cc.uoi.grAbstract: We discuss proton decay induced by dimension{5 operators in supersymmetric modelscontaining extra hypercharge{1=3 colour{triplets. We derive a general formula relating dimension{5operator to the colour{triplet mass matrix. We show that certain zeros in the triplet mass{matrixtogether with some triplet coupling selection rules can lead to elimination of dimension{5 operators.We apply this mechanism to SU(5) and flipped SU(5) theories with extended Higgs sectors.1. IntroductionProton decay is a generic feature of any unica-tion scheme since the unication of quarks andleptons in a common multiplet introduces extrainteractions that violate baryon number. Pro-ton decay rates and modes are a prediction ofGUT models that play a crucial role in theirphenomenological viability. In fact, proton de-cay has turned out to be the nemesis of manyGUT and Superstring models. It is a welcomeprediction that can be used to test GUTs. Insupersymmetric GUTs with conserved R{paritythe dominant baryon number violating operatorsare dimension D = 5, while D = 6 operatorsare in general suppressed due to the increase ofthe unication scale in comparison to its non{supersymmetric values. D = 5 operators areproportional to the Yukawa couplings and to theinverse of the heavy mass [1]. In minimal mod-els the Yukawa couplings involved are associatedwith the fermion masses. The values of these cou-plings play an important role in the nal valueof the proton decay rate and the resulting hierar-chy of existing modes. Nevertheless, Superstringembeddable models [2] or models of phenomeno-logically oriented GUTs that treat the fermionmass problem [3], come out with an extendedHiggs sector.In this talk we summarize the results of a re-cent work [4], where we propose a mechanism foreliminating or suppressing such operators basedon the use of textures of the hypercharge 1/3mass{matrix accompanied by certain constraintsof the extra triplet coupling to matter.2. Proton Decay in minimal SU(5)modelsLet us consider unied models with the minimalHiggs content to allow the beaking of the SU(5)symmetry to SU(3)c  SU(2)L  U(1)Y at theGUT scale MGUT .Non Supersymmetric SU(5) models predictproton decay as a consequence of gauge inter-actions of the heavy particles, originated whenthe SU(5) symmetry is broken, and quarks andleptons. The baryon-number-vioalting operatorsare D = 6 and they are suppressed as the squareof the mass of the heavy particles ( MGUT ).The dominant proton decay mode in these mod-els is p ! e+0. The calculated lifetime [5] is:(p! e+0)  ( MGUT3:5  104GeV)4  10311yr(2.1)While the experimental bound for this pro-cess is [6](p! e+0) > 5:5 1032yearsSince tipical values for the cuasi unicationin non-susy SU(3)cSU(2)LU(1)Y areMGUT European Network on Physics beyond the Standard Model, 1999 Mario E. Gomez1013 − 1014Gev, proton lifetime predictions ex-ceeds the experimental bounds.In SUSY SU(5) models, the scale of unica-tion is increased to MGUT  1016GeV , this isenought to bring the prediction for D = 6 medi-ated proton decay to a safe limit:(p! e+0)  10351yearsHowever proton decay is predicted, at smallerrates, due to Yukawa interactions. In this casethe supersymmetric partners of the colored tripletHiggs bosons interact with leptons (sleptons) andquarks (squarks) elds. D = 5 operators arisessuppresed only by one power of MGUT .Color triplets are contained in Higgs penta-plets h; h. The quarks and leptons are assignedto (5) +  (10) representations of SU(5). Thepart of the superpotential related to dimension{5decay will beY uij  i jh1 + Ydij  ijh1 + hh+ hh ; (2.2)where the symbol  stands for the adjoint Higgssupereld in the 24 representation.The SU(5) symmetry is broken down to theMSSM, when  gets a VEV, V, along the 24{direction. The isodoublet and colour{triplet massesareM2 = − 3V ; (2.3)M3 = + 2V (2.4)The triplets are heavy, M3  MGUT , while thedoublet pair must remain lightM2  mw. Hencea ne{tuning condition must be imposed in theparameters of the superpotential (2.2).The eective SU(3)c  SU(2)L  U(1)Y su-perpotential describing the couplings of quarksand leptons to the extra coloured triplets of theD{quark typeY uijQiQjD + YdijQiLjD + YuijEciUcjD ; (2.5)D = 5 operators can be converted into four{fermion operators by the appropiate gaugino dress-ing. Assuming roughly an overall universal su-persymmetry breaking scalemS ,the correspond-ing four{fermion operator will involve: (M3)−1mS− 4mS(M3)−3 log (M3)mS(2.6)Where  contain a combination of Yukawaand gauge couplings.The theoretical predictions [5] for the modep ! K are comparable to the experimentalbound for this mode [6](p! K) > 5:5 1032yr:Hence, the parameter space for the minimalSUSY{SU(5) is very restricted.In SU(5){models with a non-minimal con-tent of Higgs multiplets, M−13 of eq. (2.6) will bereplaced by a matrix, and therefore its null ele-ments will play an important role in the suppre-sion of D = 5{operator mediated proton decay.In the minimal flipped SU(5) U(1) model[7] matter elds come in the representationsFi(10; 1=2) ; fci (5;−3=2) ; lci (1; 5=2) (2.7)while Higgses inh(5;−1) ; h(5; 1) ; (2.8)and inFh(10; 1=2) ; Fh(10;−1=2) : (2.9)The part of the superpotential relevant forthe beaking of the uniying symmetry and Yukawaterms is:fijFi Fj h+ yijFi fcj h+ rij lci fcj h+hh+ FhFhh+ FhFhh (2.10)VEV’s of Fh and Fh along the neutrino-likecomponent break the SU(5)U(1) symmetry tothe MSSM. A great advantage of the \flipped"SU(5) model over the ordinary one is that ofthe realization of the \triplet{doublet splitting"mechanism without ne{tuning the parametersof the superpotential (2.10). In this case the dou-blet mass is given by the parameter  which canbe  mw while the mass matices for the triplets:M3 =0 VV 0(2.11)Where the entry 22 is null since the pair Fh Fhhas to be massless in order to realize the SU(5)U(1) breaking to the standard model. Since inthis model there is not DD mass term, D = 5-operators are naturally suppresed.2European Network on Physics beyond the Standard Model, 1999 Mario E. Gomez3. How to suppress dimension{5 op-erators in eective models withextra tripletsLet us consider a general supersymmetric modelcontaining some extra hypercharge{1=3 colour{triplets 1. The eective SU(3)cSU(2)LU(1)Ysuperpotential describing the couplings of quarksand leptons to the extra coloured triplets of theD{quark typefijQiQjD + yijQiLjD + rijEciUcjD ; (3.1)where i; j = 1; 2; 3 are the usual generation in-dices and  = 1; :::; N is an extra index describ-ing the multiplicity of triplets and repeated in-dices are summed. In addition the eective tripletmass matrix will be of the form(M3)DD (3.2)whereM3 is in general non{diagonal.We can always go to a basis in which thetriplet mass{matrix is diagonalD = SD0 ; D = UD0 (3.3)M3D  diag(m1;m2;    ;mN) = STM3 U(3.4)where the matrices S and U are unitary. In thisbasis we can easily evaluate D = 5 operatorsresulting from Higgs triplet fermion exchange,and then recast the result in the original basis.Assuming that all triplets are massive (mi 6=0 ; i = 1; : : : ; N), operators with the structureQiQj Qk Ln will be proportional to2OQQQLijkl =NX;;γ=1fijSγ(M3−1D )γUγykn=NX;=1fij(M3−1)Tykn=1det(M3)NX;=1fij cof(M3) ykn(3.5)1This superpotential arises in the case of the stan-dard SU(5) with extra Higgs 5{plets or from the flippedSU(5)U(1) with both extra Higgs 5{plets and 10{plets.2The corresponding four{fermion operator, assum-ing roughly an overall universal supersymmetry breakingscale mS , will involve(M3)−1mS− 4mS(M3)−3 log (M3)mS .Analogous formulas hold for D = 5 operators ofthe type QiQjUckEck.Suppose now that we want to eliminate alldimension ve operators. Assuming that the Yukawacouplings fij and yij are in general unrelatedand detM3 6= 0 , equation (3.5) implies thatthe necessary and sucient condition for vanish-ing of the OQQQLijkl operator is that for every pairof triplets (D,D, ;  = 1; : : : ; N) that docouple to quarks and leptons respectively (fij 6=0 and hij 6= 0) the cofactor of the correspondingtriplet mass matrix element (M3) vanishes3OQQQLijkl = 0() cof(M3) = 08 (; ) 2  = f(; ) : fij 6= 0and hkl 6= 0g : (3.6)It is obvious that in the case where all triplets(D’s and D’s) couple to matter the suppressionof dimension ve operators (3.5) is not possiblesince (3.6) leads to det(M3) = 0. Nevertheless,if for some reason (discrete symmetry, R{parity,anomalous U(1), accidental symmetry) some ofthe fij and/or ykl are zero and the triplet massmatrix is such that the cofactors of the appropri-ate matrix elements are zero then the associateddimension{5 operator vanishes.The previous discussion leads to the possi-bility that in a model with extra D{quark tripletsdimension{5 operators can be eliminated by usingtextures of triplet mass matrices and the triplet{matter couplings.To be concrete let us give a simple exampleof such an eective theory. Consider the case ofan eective theory with two extra triplets. Onlythe rst couples to matter through the superpo-tential termsf1ijQiQjD1 + y1ijQiLjD1 + r1ijEciUcjD1 (3.7)and their mass{matrix has the formM3 =11 1221 0: (3.8)Since f2ij = y2ij = 0 evaluation of (3.5) leads toOQQQLijkl = f1ij cof(M3)11 y1kn  cof(M3)11 = 0(3.9)3We consider here the triplet D as coupled to quarksand leptons if at least one fij 6= 0 and similarly for anti{triplets.3European Network on Physics beyond the Standard Model, 1999 Mario E. GomezIt is remarkable that if we remove the secondpair of triplets of the model (which do not coupledirectly to matter) the usual dimension{5 oper-ators reappear. We shall study below that thisnice property can be incorporated in SU(5) mod-els.4. SU(5)models without dimension{5 operatorsLet us consider now an SU(5) model with twopairs of Higgs pentaplets h; h ;  = 1; 2 of whichonly the rst couples to matter. The quarks andleptons are assigned to (5) +  (10) representa-tions of SU(5). The part of the superpotentialrelated to dimension{5 decay will befij  i jh1 + yij  ijh1 +2X;=1(hh + hh) ; (4.1)where the symbol  stands for the adjoint Higgssupereld in the 24 representation. The isodou-blet and colour{triplet mass matrices are corre-spondingly of the formM2 = − 3V ; (4.2)M3 = + 2V (4.3)The well known ne{tuning that guarantees amassless pair of isodoublets amounts todet(M2) = 0 : (4.4)The proton decay rate through D = 5 operatorsis, according to equation (3.5), determined by thecofactor of the 1− 1 element of the triplet massmatrixcof(M3)11 = (22 + 222V ) : (4.5)Hence, choosing 22 = −222V dimension{5 op-erators vanish. This condition is perfectly com-patible with the previous ne{tuning condition(4.4). It is very interesting that proton decaythrough D = 5 operators can be set to zerothrough a condition on the couplings 4.4Of course, proton decay still goes through at the (sup-pressed) rate of D = 6 operators.In the framework of our standard SU(5) ex-ample the required zero in the inverse triplet massmatrix does not correspond to any symmetry andis in a sense a second ne{tuning. Neverthe-less, the general conclusion is that zeros of thetriplet mass matrix, perhaps attributable to sym-metries, can stabilize the proton.The superpotential considered above in (4.1)is not the most general one. In fact, the casethat all 5{plets couple to matter cannot be re-duced to (4.1) since it would require a dierentHiggs 5{plet rotation for each generation of mat-ter. However, we should emphasize the fact thatin SU(5) models with non minimal Higgs con-tent, the constraints imposed by proton decayon the parameter space and triplet masses canbe relaxed.5. Dimension{5 operators in exten-sions of the flipped SU(5)In spite of the nice features of the minimal flippedSU(5) model, all attempts to obtain such a modelfrom strings have yielded up to now non{minimalmodels. Such models include(a) extra pairs of low energy Higges (h, h) and/or(b) extra pairs of SU(5) U(1) breaking Higges(Fh, Fh).We are thus motivated to study the presenceof dimension{5 operators in such models. As weshall see contrary to the minimal case, such ex-tensions of the flipped SU(5) model are not au-tomatically free of dimension{5 operators.The relevant part of the superpotential as-suming N5 pairs of Higgs 5{plets (h; h ;  =1; : : : ; N5) that couple to matter and N10 pairs ofHiggs 10{plets (Fh; FhA ; A = 1; : : : ; N10) thatdo not couple to matter, will have the formfijFi Fj h + yijFi fcj h +rij lci fcj h + hh +mABFhAFhB+ ABγFhAFhBhγ + ABγFhAFhBhγ(5.1)where A;B = 1;    ; N10 , ; ; γ = 1;    ; N5 .Assuming GUT symmetry breaking to an arbi-trary direction in the Higgs 10{plet space((V1; V2; : : : ; VN10) and similarly for bars)5, we5D{flatness requiresPAV 2A =PAV A24European Network on Physics beyond the Standard Model, 1999 Mario E. Gomezobtain the triplet mass matrix 6M3 = vAvA mAB(5.2)where  is the doublet mass{matrix and vA =2ABVB , vA = 2ABV BF{flatness demands det(m) = 0 in order tohave at least one pair of massless Higgs decupletswhich will realize the GUT symmetry breaking.One can actually choose m to have only one zeroeigenvalue so that all remnants of the Higgs de-cuplets will become heavy.Let us now start our study by a simple exam-ple. Consider the flipped model with two pairs ofHiggs 5{plets and one pair of Higgs 10{plets. As-suming for simplicity that the 5-plet mass matrixis diagonal, the explicit form of the triplet matrixis 7M3 =0@ 0 0 1V0  2V1V 2V 01A (5.3)and det(M3) = 11V The transpose of inversetriplet matrix entering in formula (3.5) is(M−13 T =0BB@2211− 21− 211  1CCA (5.4)where the dots stand for elements which are ir-relevant. It is now obvious that in this modeldimension ve operators cannot be eliminatedsince even in the case 2 = 2 = 0 the 22 ele-ment does not vanish. If we want to eliminatethem we have two solutions :(i) assume that the extra pair of 5{plets does notcouple to matter. In this case only the 11 elementof the matrix in (5.4) is relevant and it vanishesfor 2 = 0 (or 2 = 0).(ii) make the milder assumption that one of the5{plets (e.g. h2) does not couple to the up quarks(or similarly h2 does not couple to the down). Inthis case the second column (or line) of the ma-trix in (5.4) becomes irrelevant and the column(or line) left vanishes for 2 = 0 (or 2 = 0).6In a (D1;    ;DN5 ; (dcH )1;    ; (dcH)N10 ) versus(D1;    ; DN5 ; (dcH)1;    ; (dcH)N10 ) basis, where withD we denote the triplets which lie inside the Higgs5{plets and with DH the triplets that lie inside the Higgs10{plets.7We have renamed 1 = 111 ; 2 = 112.Another case that could arise is the existenceof extra decuplets. The simplest of these cases isfor N5 = 1 and N10 = n  2.cof(M3)11 = detmdetM3 (5.5)which means that proton decay is absent only inthe case that the restricted mass{matrix of thetriplets not coupled to matter hasdetm = 0 (5.6)This constrain naturally arises in the context ofthe flipped SU(5)  U(1) model as consequenceof F{flatness as we mentoned above.In the more general case where N5 and N10are arbitrary dimension{5 operators can be sup-pressed only in the case N5  N10 . Furthermoreone has to require that the Higgs decuplet massmatrix has N5 zero eigenvalues. This is compat-ible with symmetry breaking and with the re-quirement of making all triplets heavy but leavesN5−1 pairs of Q(3;2; 1=6)+Q(3;2;−1=6) mass-less. This feature does not necessarily mean thatthis possibility is ruled out. On the contrary onecan consider the cases where extra Q’s have inter-mediate masses which are small enough to su-ciently suppress dimension{5 operators but theyare still compatible with renormalization grouprequirements. The appearance of extra vector{like pairs of Q and D type multiplets with in-termediate masses is a welcomed feature in thecontext of flipped SU(5)U(1) models that raisethe unication scale to the string scale [8].6. ConclusionsOur main result is that textured zeros of thecolor{triplet mass{matrix as well as Yukawa se-lection rules can eliminate certain dimension{5operators. In order to be specic we focused onSU(5) models. In particular, we showed that in-troducing an extra pair of Higgs pentaplets in thestandard supersymmetric SU(5) , with speciccouplings, can eliminate these operators. We alsoconsidered the case of the flipped{SU(5) modelwith extra pentaplets and decuplets and ana-lyzed the conditions for vanishing proton decaythrough dimension{5 operators. Flipped{SU(5)with extra decuplets was shown to be D = 55European Network on Physics beyond the Standard Model, 1999 Mario E. Gomezoperator{free as it happens in the case of theminimal model. However, flipped{SU(5) withextra Higgs pentaplets is not automatically freeof dimension{5 operators. We have proposed asolution to this problem which involves a tex-ture of the pentaplet matrix together with cer-tain constraints on the pentaplet couplings tomatter.AcknowledgmentsThis work is supported by E.U. under theTMR contract \Beyond the Standard Model",ERBFMRX-CT96-0090.References[1] S. Weinberg, Phys. Rev. D26 (1982) 287;N. Sakai and T. Yanakida, Nucl. Phys. B197(1982) 533.[2] see e.g.I. Antoniadis and G. Leontaris, Phys. Lett.B216 (1989) 333;S. Ranfone and J. W. F. Valle, Phys. Lett. B386(1996) 151;J. Rizos and K. Tamvakis, Phys. Lett. B414(1997) 277.[3] see e.g. K. S. Babu and S. M. Barr, Phys. Rev.Lett. 75 (1995) 2088.[4] M. E. Gomez, J. Rizos and K. Tamvakis Phys.Rev. D59 (1998)015015.[5] P. Nath and R. Arnowitt, hep-ph/9808465 andreferences therein.[6] C. Caso it et al., The European Physics JournalC3(1998)1.[7] S. M. Barr, Phys. Lett. B112 (1982) 219;J. P. Derendinger, J. H. Kim, D. V. Nanopoulos,Phys. Lett. B194 (1987) 231.[8] J. L. Lopez and D. V. Nanopoulos, Nucl. Phys.B399 (654) 1993;see also J. L. Lopez, D. V. Nanopoulos and A.Zichichi, hep-ph 9307211 and references therein.6

Yukawa couplings and proton decay in SUSY models

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