We derive the scalar mass matrices in effective supergravity models augmented
by a $U(1)_F$ family symmetry. Simple relations between $U(1)_F$ charges and
modular weights of the superfields are derived and used to express the matrices
with a minimum number of parameters. The model predicts a branching ratio for
the $\mu\to e\gamma$ process close to the present experimental limits.Comment: LaTex file, 11 pages, 1 Figure.Typos corrected and a reference has
  been added. Accepted in Phys. Lett. 

Leontaris, G. K.

Tracas, N. D.

Physics Letters B

English

arXiv

We derive the scalar mass matrices in effective supergravity models augmented by a $U(1)_F$ family symmetry. Simple relations between $U(1)_F$ charges and modular weights of the superfields are derived and used to express the matrices with a minimum number of parameters. The model predicts a branching ratio for the $\mu\to e\gamma$ process close to the present experimental limits.We derive the scalar mass matrices in effective supergravity models augmented by a $U(1)_F$ family symmetry. Simple relations between $U(1)_F$ charges and modular weights of the superfields are derived and used to express the matrices with a minimum number of parameters. The model predicts a branching ratio for the $\mu\to e\gamma$ process close to the present experimental limits.We derive the scalar mass matrices in effective supergravity models with the standard gauge group augmented by a U (1) F family symmetry. Simple relations between U (1) F charges and modular weights of the superfields are derived and used to express the matrices with a minimum number of parameters. The model predicts a branching ratio for the μ → eγ process close to the present experimental limits

Leontaris, G.K.

Tracas, N.D.

CERN Document Server

NTUA-65/97IOA-TH/97-005CERN-TH/97-268Modular Weights, U(1)’s and Mass Matrices?G.K. Leontaris a;b and N.D. Tracas c;baTheoretical Physics Division, Ioannina University, GR-451 10 Ioannina, Greeceb CERN, Theory Division, Genevac Physics Department, National Technical University, 157 80 Athens, Greece.AbstractWe derive the scalar mass matrices in eective supergravity models with the standardgauge group augmented by a U(1)F family symmetry. Simple relations between U(1)Fcharges and modular weights of the superelds are derived and used to express thematrices with a minimum number of parameters. The model predicts a branchingratio for the ! eγ process close to the present experimental limits.?Research supported in part by TMR contract ERBFMRX-CT96-0090, ENE-1170/95 andENE-15815/95September 1997.{2{The Minimal Supersymmetric Standard Model (MSSM) emerges as the most naturalextension of the Standard Model (SM) in the context of the unication of all interactions.Although supersymmetric models solve the hierarchy problem, the plethora of arbitraryparameters requires a further step beyond the MSSM. The N = 1 supergravity coupled tomatter stands promising [1]. Yet, there are many essential parameters (Yukawa couplings,content of the chiral multiplets, etc.) to be chosen by the model builder. In this scene,string theory appears the only known candidate theory that can in principle predict allthe required parameters. String theory puts rather strong constraints on many of theparameters of the resulting N = 1 eective supergravity, which appears as its low-energylimit. Thus, the kinetic terms must have a certain structure, the Lagrangian should obeythe string duality symmetries, while several constraints are imposed on the superpotentialand the Yukawa couplings [2].The subject of this letter is to reproduce the observed family hierarchy of the fermionmasses and moreover to predict the corresponding mass matrices in the scalar supersymmet-ric sector. This is done in the context of residual stringy U(1) symmetries [3] left from thelarge gauge group at a high scale. In particular, combining modular invariance constraintsand U(1) invariance of the superpotential, the scalar mass matrices are given in terms ofpowers of an expansion parameter hi=M , where hi is the vacuum expectation value of asinglet eld and M is a high (string) scale. These powers are written in terms of modularweight dierences. Further, the consequences in the lepton flavour non-conserving reaction! eγ are examined. Its branching ratio is found close to the present experimental limits.We start with a quick review of the N = 1 supergravity, which introduces a real gauge-invariant Ka¨hler function with the general form [4]G(z; z) = K(z; z) + log j W(z) j2 (1)where K(z; z) is the Ka¨hler potential and W(z) is the superpotential. Denoting (; Q) byz, where  stands for the dilaton eld S and other moduli Ti, while Q represents the chiral{3{superelds, the Ka¨hler potential at tree level can be written as followsK(; ; Q; Q) = − log(S + S)− nhn log(Tn + Tn) + Zij(; ) Qie2VQj +    (2)The superpotential W(z) is a holomorphic function of the chiral superelds Qi and at thetree level is given byW(; Q) =13ijk()QiQjQk +12ij()QiQj +    (3)In both (2) and (3),    stand for possible non-renormalizable contributions. Terms bilinearin the elds Qi refer in fact to an eective Higgs mixing term.Now under the modular symmetries, the moduli transform as T ! (aT − {b)=({cT + d),where a; b; c; d constitute the entries of the SL(2;Z) group elements with a; b; c; d 2 Z andad− bc = 1. These imply the following transformation rules [5]Qi ! QiYktnkik ; Zij ! ZijYk(tk)−nki (tk)−nkj ; W !YktnkWk W; (4)where we have introduced the notation tk = {ckTk + dk. The exponent nki is the modularweight of Qi with respect to the modulus Tk.Let us now introduce into the Ka¨hler function non-renormalizable terms through twoelds,  and , which are singlets under the low energy standard gauge group, while theycarry charges q = −q under the U(1)F family group. The lower order (in Qi’s) non-renormalizable terms can be written in the formKijrij hiM1!rijQi Qj +Kij~rij hiM2!~rijQi Qj : (5)These terms should be invariant under the U(1)F symmetry. Assigning U(1)F charges qifor the matter elds one getsqi + qj + qrij = 0; qi + qj + q~rij = 0: (6)Similar non-renormalizable terms could also appear in the superpotential.{4{After this short review we come to the mass matrix textures. The SU(2)L invariance,together with the requirement to have symmetric mass matrices, leads us to assign thesame U(1)F charge to all quark members of the same family qi, while the same should beapplied to the leptons of the same family li. The full anomaly-free Abelian group involvesan additional family-independent component and with this freedom we may make U(1)Ftraceless without any loss of generality. Thus q1 + q2 + q3 = 0 and l1 + l2 + l3 = 0.If the light Higgs H2, responsible for the masses of the up quarks, and H1, responsiblefor the down quarks and leptons have U(1)F charge, so that only the (3,3) renormalizableYukawa couplings to H2 and H1 are allowed, namely2q3 + h2 = 0; and 2l3 + h1 = 0; (7)only the (3,3) element of the associated mass matrix will be non-zero. The remainingentries are generated when the U(1)F symmetry is broken. A straightforward consequenceof this fact is the equality of the two Higgs U(1)F charges (h1 = h2), since H1 provides alsothe mass to the bottom quark while we have assumed equal U(1)F charges within a family.A general non-renormalizable relevant term in the superpotential is of the formYijQiucjH2 M!xij: (8)Owing to U(1)F invariance of the superpotential, we have the constraintqi + qj + h2 + xijq = 0 (9)and similarly for the parameter  terms. The allowed powers of non-renormalizable termsin each entry are determined by the charges (q = −q)xij = jq2 − q3qj0BBBBBB@2ja+ 1j jaj ja+ 1jjaj 2 1ja+ 1j 1 01CCCCCCA (10){5{where a = 3q3=(q2−q3), and we have used the condition (7). Suppressing unknown Yukawacouplings Yij and their phases, which are all expected to be of order 1, we arrive at thefollowing mass matricesmU 0BBBBBB@2ja+1j jaj ja+1jjaj 2 ja+1j  11CCCCCCAmt (11)mD 0BBBBBB@~2ja+1j ~jaj ~ja+1j~jaj ~2 ~~ja+1j ~ 11CCCCCCAmb (12)where ~ = ( hiM1)j(q2−q3)=q j,  = ( hiM2)j(q2−q3)=qj (M1 and M2 being two high scales). Thecharged lepton mass matrix may similarly be determined. The equality h1 = h2, togetherwith (7) has also the consequence q3 = l3, which implies the successful relation mb = mat unication. We then getmL 0BBBBBB@~2ja+bj ~jaj ~ja+bj~jaj ~2jbj ~jbj~ja+bj ~jbj 11CCCCCCAm (13)where b = (l2 − q3)=(q2 − q3).The powers of the above matrices can be written in terms of the modular weightsas follows. As we have already discussed in the introduction, the superpotential trans-forms covariantly under the modular symmetry. Let us denote by nQi ; nui; ndi ; nh2; nthe modular weights for the corresponding elds with respect to a certain modulus. Forthe non-renormalizable term of the form (8), the modular weights obey the equationnQi + nuj + nh2 + xijn = nW . Combining this relation with the U(1)F invariance andthe fact thatP3i=1 qi = 0, we obtain the general formulaqj =q3nXinQji =q3nXinuji =q3nXindji (14){6{where nQji = nQj − nQi and correspondingly for nuji and ndji . The third equality comesfrom the down-quark mass matrix non renormalizable contributions corresponding to aterm like (8). Similar relations hold for the lepton modular weights. Using the aboverelation, we may obtain an elegant form of the matrix (10), which expresses the powers ofthe allowed non-renormalizable entries only in terms of modular weight dierences [6]. Weobtainxij =1n0BBBBBB@2nQ31 nQ31 + nQ32 nQ31nQ31 + nQ32 2nQ32 nQ32nQ31 nQ32 01CCCCCCA (15)The positivity of the entries requires the conditions nQ31n > 0 and nQ32n > 0. We canalso express the powers of the matrix (12) in terms of modular weight dierence. This iseasily done by expressing the parameter a in the forma =nQ13 + nQ23nQ23: (16)From (15) we conclude that the hierarchical fermion mass spectrum requires all three nQi’sto be dierent. Models with equal nQi ’s, but dierent qi’s (necessary to create hierarchy),require q = 0. In this case the U(1)F charges are not related to the modular weights andthe constraint (14) does not hold.We next turn to the lepton fermion mass matrix. The phenomenological constraintl3 = q3 imposes the following relation on the modular weights of the quark and leptongenerationsnL13 − nQ13 = nQ23 − nL23  : (17)As a result, the U(1)F structure permits to express the powers yij of the lepton termLiejH10B@(−)M1CAyij(18){7{by the following matrixyij =1n0BBBBBB@2(nQ13 + ) nQ13 + nQ23 nQ13 + nQ13 + nQ23 2(nQ23 − ) nQ23 − nQ13 +  nQ23 −  01CCCCCCA (19)whilst the corresponding constraints for the positivity of the entries are nL13n > 0 andnL23n > 0. The powers of the matrix (13) can also be expressed in the same way bywriting b in the formb =nQ23 − nQ23: (20)We now turn to the scalar part. At the tree level the scalar mass matrices receivecontributions only along the diagonal, since terms of the form / QiQi have zero U(1)Fcharge. Using powers of the elds ;  scaled by the M , we may ll in the remaining entries.It can easily be seen that the allowed U(1)F structure of the powers in the scalar mass termis1n0BBBBBB@0 nQ12 nQ13nQ12 0 nQ23nQ13 nQ23 01CCCCCCA (21)Thus, the powers of the parameters hi; hi are simply determined by the dierences nQijfor the squark matrix and similarly for the sleptons (remember that since the U(1)F chargeis the same within a family, (14) tells us that nQij = nuij = ndij ).Using again the parameters a and b entered in the fermion mass matrices, we can expressthe squark mass matrix in the formm2~Q 0BBBBBB@1 ja+6j ja+1jja+6j 1 ja+1j  11CCCCCCAm23=2 (22){8{where m3=2 is the gravitino mass. Similarly, for the sleptons we obtainm2~lL;R 0BBBBBB@1 ~ja+6bj ~ja+bj~ja+6bj 1 ~jbj~ja+bj ~jbj 11CCCCCCAm23=2 (23)Obviously in the case of b = 1 the two matrices are identical since this case corresponds toequal U(1)F charges in the quark and the leptonic sector, li = qi. In fact, it can be checkedthat the phenomenological analysis of the fermion mass spectrum allows two values of b,namely b = 1 or 1=2 [7], while   0:053 and ~  0:23.The above results show that U(1)F symmetries necessarily lead to low energy modelswhere the Yukawa and its corresponding scalar mass matrices are not simultaneously diag-onalized. As a result, flavour violation is possible and in general one should check whethersuch models can pass also the flavour violation tests. One of the most popular flavournon-conserving processes is the ! eγ decay. We have calculated the branching ratio forthis process in order to compare it with the present experimental limits. This calculationrequires the diagonalization of the 6 6 scalar mass matrix~M2~l =0BB@ m2~lL AlhH1i+mL tan(AlhH1i+mL tan)y m2~lR1CCA (24)Here, as usual, Al is the trilinear parameter entering the scalar potential,  is the Higgsmixing term and tan  is the Higgs vev ratio. Since lepton mass matrices are symmetric,left and right diagonalizing matrices coincide. Further, due to the properties of the U(1)Fsymmetry of the model, left (m2~lL) and right (m2~lR) scalar mass matrices are the same.Moreover, here we restrict on the case of small tan  regime, where the chirality changingdiagrams are suppressed. In the general case, of course, and in the large tan scenario,they become important. We have considered contributions from one loop graphs involvingneutralino-charged slepton or chargino-sneutrino states in the loop. The diagrams of thisprocess are shown in Fig. 1. We have diagonalized the lepton and the slepton mass matri-ces and found the corresponding amplitude for each neutralino/chargino graph. Then by{9{eµNe~iγµ eν~iχγFigure 1: The ! eγ decay via supersymmetric graphs.diagonalizing the Wino mass matrices we evaluated the total amplitude and the branchingratio. For sensible values of m3=2  m1=2  O(mW ) (initial values for the scalar and gaug-ino masses respectively) and standard GUT initial conditions for gauge couplings, the valueof the BR!eγ can reach the order of 10−12. Thus, this rare decay gives the opportunity totest the viability of the above U(1)F -like model in future experiments.In conclusion, we have considered the scalar mass matrices in supergravity models withthe standard SU(3)SU(2)LU(1)Y gauge group augmented by a U(1)F family symmetry.Using modular invariance of the Ka¨hler potential and the superpotential, we have derivedcertain relations between U(1)F charges and the modular weights of the elds. As a result,the scalar mass matrix entries are found to depend only on certain powers, which areproportional to the dierence of modular weights. We have calculated, as an example, theprocess  ! eγ, which is found, for a wide range of the parameter space (tan , m3=2,m1=2), to be very close to the present experimental limits. 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Modular Weights, U(1)'s and Mass Matrices

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