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IFT-2004/31 CERN-PH-TH/2004-263 February 2, 2008 Update on Fermion Mass Models with an U(1) Anomalous Horizontal Symmetry 5 0 0 2 n a a a b, J Piotr H. Chankowski , Kamila Kowalska , Stéphane Lavigna 1 and a,b 0 Stefan Pokorski 1 1 a v Institute of Theoreti al Physi s, Warsaw University, Ho»a 69, 00-681, Warsaw, Poland b 1 Theory Division, Physi s Department, CERN, CH-1211 Geneva 23, Switzerland 7 0 1 0 5 0 / h Abstra t p - p e We re onsider models of fermion masses and mixings based on a gauge anomalous horizontal U(1) h symmetry. In the simplest model with a single (cid:29)avon (cid:28)eld and horizontal harges of the : v same sign for all Standard Model (cid:28)elds, only very few harge assignements are allowed when i X all experimental data, in luding neutrino os illation data, is taken into a ount. We show that r a a pre ise des ription of the observed fermion masses and mixing angles aUn(1ea)sily be obtained by generating sets of the order one parameters left un onstrained by the symmetry. The orresponding Yukawa matri es show several interesting features whi h may be important for (cid:29)avour hanging neutral urrents and CP violation e(cid:27)e ts in supersymmetri models. 1 Permanent address: Servi e de Physique Théorique, CEA-Sa lay, F-91191Gif-sur-Yvette Cedex, Fran e. 1 Introdu tion The origin of the fermion mass and mixing textures is a hallenge for physi s beyond the Standard Model. One promising approa h to that problem is based on hypotheti al horizontal symmetries whi h are spontaneously broken by va uum expe tation values of some (cid:16)(cid:29)avon(cid:17) Φ (cid:28)elds . Hierar hi al patterns in the fermion mass matri es( Φan/Mth)enn be expMlained by the Froggatt-Nielsen me hanism [1℄, as due to suppression fa tors h i , where is the s ale n of integrated out physi s and the power depends on the horizontal group harges of the fermion, Higgs and (cid:29)avon (cid:28)elds. Fermion mass models based on abelian [2℄(cid:21)[10℄ and non-abelian [11℄ horizontal symmetries have been widely dis usssed in the literature. Ea h approa h has its own virtues and short om- U(2) ings, parti ularly when realized in supersymmetri models, whi h we onsider here. and SU(3) symmetries are quite predi tive for fermion mass matri es on e the pattern of symmetry breaking is spe i(cid:28)ed. However, the quantitative des ription of fermion masses and mixings U(1) requires a rather ompli ated stru ture of symmetry breaking. The main short oming of horizontal symmetries as models of fermion masses is the dependen e of the quantitative predi tions on arbitrary order one oe(cid:30) ients, the remnant of the integrated out unknown physi s. In prin iple also the arbitrariness in the hoi e of the abelian harges for fermions looks less U(2) SU(3) appealing than the rigid stru tures of the essentially unique hoi es of and as on- tinuous horizontal non-abelian symmetry groups. But that last point is balan ed by the fa t U(1) that the breaking of the symmetry is mu h simpler. U(1) gaugegroupfa torsaregeneri instringmodels. Oneparti ularlysimpleandattra tive U(1) horizontal symmetry is the anomalous often found in heteroti string ompa ti(cid:28) ations. Anomaly an ellation by the Green-S hwarz me hanism [12℄ is possible only under ertain U(1) onditionswhi h strongly onstrain the possible hoi es of horizontal harges for fermions. It has already been noti ed long ago that those onstraints are at a qualitative level amazingly onsistent with the quark masses and mixings [4℄(cid:21)[9℄. A parti ularly appealing feature of the U(1) Φ /M anomalous is that the symmetry breaking parameter h i an be omputed in terms of the horizontal harges, and that its value in expli it models turns out to be very lose to the Cabibbo angle. Furthermore, with positive harges for matter (cid:28)elds and negative for the (cid:29)avon (cid:28)eld (or vi e-versa), this va uum is unique. The purpose of this note is to update the predi tions for fermion masses of the simplest, U(1) Φ string-inspired gauge anomalous models with a single (cid:29)avon (cid:28)eld . We are motivated by several fa tors. One is the re ent experimental progress in neutrino physi s. Neutrino masses U(1) and mixings an be well des ribed by models based on horizontal symmetries, and it is therefore interesting to he k what phenomenologi al onstraints, in addition to the Green- S hwarz anomaly an ellation onditions, are put on the horizontal harges by the requirement U(1) that the same gauged symmetry explains both the quark and lepton mass hierar hies. In fa t, assuming that all Standard Model (cid:28)elds have horizontal harges of the same sign, we (cid:28)nd a very limited number of possible harge assignments. Furthermore, it is interesting to he k the su ess of these models beyond the qualitative 1 level. In this paper, we perform a omplete numeri al (cid:28)t of a few representative models to fermion masses and mixings, with the in lusion of omplex order one oe(cid:30) ients and with tanβ attention paid to the dependen e. Finally, in supersymmetri models there are new sour es of (cid:29)avour and CP violation indu ed by virtual sfermion ex hanges. Sfermion mass matri es depend both on the pattern of supersymmetry breaking and on the rotations to the super-CKM basis whi h are determined by the fermion mass matri es. Thus, it is useful to have an expli it set of su essful fermion mass models, de(cid:28)ned by a set of horizontal harges and omplex order one oe(cid:30) ients. The fa t that only a small number of harge assignments are allowed makes it possible to test the phenomenologi al impli ations of an anomalous gauge horizontal symmetry. In a forth oming paper, we shall use this set of models to study (cid:29)avour hanging neutral urrents (FCNCs) in D supersymmetry breaking s enarios with dominant -term breaking. The paper is organized as follows. In the next se tion, we review the basi properties of U(1) fermion mass models based on an anomalous and list the phenomenologi ally allowed harge assignements. In se tion 3 we generate sets of omplex order one oe(cid:30) ients giving a pre ise des ription of fermion masses and mixings, and study the features of the asso iated Yukawa matri es relevant for FCNC and CP violating pro esses. Finally, we present our on- lusions in se tion 4. U(1) X 2 Constraints on harges R The models we onsider in this paper are extensions of the ( -paritˆy onserving) MSSM with a U(1) Φ X X horizontal gauge abelian symmetry and a hiral super(cid:28)eld with - harge normalized 1 X to − , whose va uum eQˆxpe UtâctioDnˆcvalLûe bErêcaksAth=e h1o,r2i,z3ontaHˆl symmeHtˆry. The - hargesqof the M¯ SSM super(cid:28)elds A, A, A, A, A ( ), u and d are denoted by A, u¯ d l e¯ h h A A A A u d , , , Nˆc, and , resn¯pe tively. The model also ontains three right-handed neutrino super(cid:28)elds A with harges A, whi h are needed to generate the neutrino masses via the seesaw me hanism [13℄. No additional matter harged under the SM gauge group is assumed, although the forth oming analysis would not be altered by the presen e of ve tor-like matter U(1) X under both the SM gauge group and . The Yukawa ouplings of quarks and leptons are generated via the Froggatt-Nielsen me hanism [1℄, from nonrenormalizable superpotential terms of the form: Φˆ u¯A+qB+hu CABUˆcQˆ Hˆ . u A B u M (1) ! These terms arise upon integrating out heavy ve tor-like (super)(cid:28)elds, the so- alled Froggatt- M Nielsen (cid:28)elds, whose hara teristi mass s ale is . In string ompa ti(cid:28) ations, the rle of the M Froggatt-Nielsen (cid:28)elds is played by massive string modes, and is identi(cid:28)ed with the Plan k s ale (although ve tor-like (cid:28)elds may also be present among the massless string modes). Aftêr φ Φ breaking of the horizontal symmetry by the VEV of the s alar omponent of the super(cid:28)eld , 2 ǫ φ /M oneobtainse(cid:27)e tive Yukawa ouplingssuppressed bypowers ofthesmallparameter ≡ h i : YAB = CAB ǫu¯A+qB+hu . u u (2) CAB The fa tors u are not onstrained by the horizontal symmetry and are assumed to be of order one. Then the hierar hy of Yukawa ouplings is determined, up to these unknown fa - tors, by the harges of the MSSM (cid:28)elds. Sin e holomorphi ity of Φˆthe superpoYtenAtBia=l fo0rbids nu¯on+reqnor+mhaliz<ab0le terms witYh AaBn=eg0ativue¯ p+owqer+ofhthe super(cid:28)eld , one has u if 2 A B u u A B u . Similarly, if is not an integer. Within the above assumptions, one an show that the horizontal abelian symmetry has to be anomalous [9℄ (see also Refs. [4, 5℄). Indeed, one has the following relations: detM detM v3v3 ǫC3+3(hu+hd) , u d ∼ u d (3) detMd/detMe ǫ−21(C1+C2−83C3)+hu+hd , ∼ (4) C (2q +u¯ +d¯ ) C (3q +l )+h +h C 1 (q +8u¯ +2d¯ + where 3 ≡ A A A A , 2 ≡ A A A u d and 1 ≡ 3 A A A A 3l +6e¯ )+h +h SU(3) SU(3) U(1) SU(2) SU(2) A A u d C c X L L P are the oe(cid:30) ientPs of the mixed - - P , - - U(1) U(1) U(1) U(1) v X Y Y X u,d and - - anomalies, respe tively, and are the va uum expe tation valuesofthetwoHiggsdoublets. Ifallanomaly oe(cid:30) ientswere vanishing,thehierar hyamong h +h h +h = 6 8 ǫ = λ λ u d u d quark masses would require a large, positive value of ( − for , where tanβ is the sine of the Cabibbo angle, depending on the value of ), while the relation (4) would h +h = 1 2 ǫ = λ u d require a mu h smaller value ( or for ). Therefore, the horizontal symmetry has to be anomalous if it is to explain the observed fermion mass hierar hy. This fa t provides the main motivation for onsidering a gauged horizontal symmetry. It is well-known that abelian gauge anomalies an be ompensated for by the Green-S hwarz 3 me hanism [12℄, as is ommon in four-dimensional heteroti string ompa ti(cid:28) ations . This requires that the following relations between anomaly oe(cid:30) ients be satis(cid:28)ed: C C C C X 1 2 3 X = = = = Tr , C = 0 , XXY k k k k 12 (5) 1 2 3 X C U(1) X X X where is the oe(cid:30) ient of the ubi anomaly, Tr is the oe(cid:30) ient of the mixed U(1) k G X a a -gravitational anomaly, and is the Ka -Moody level of the gauge group , whi h 2 This on lusion an be evaded if the Froggatt-Nielsen (cid:28)elds, instead of being ve tor-like under both the U(1) U(1) X X Standard Model gauge group and the group as usually assumed, are hiralunder . In su h a ase X e(cid:27)e tive operators arrying a negative - harge an beUi(n1d)u ed in the low-energyLê(cid:27)Lê tHiˆveHˆtheory. Moreover, X A B u u even if the Froggatt-Nielsen(cid:28)elds areve tor-likeunder , e(cid:27)e tive operators with a negative X - harge may be indu ed by the seesaw me hanism, sin e right-handed neutrino super(cid:28)elds are hiral under U(1) X (see e.g. the model of Ref. [14℄). U(1) 3 Anomalous 's are also ommon in four-dimensional open string ompa ti(cid:28) ations. However, while heteroti string ompa ti(cid:28) ations ontain at most one anomalousabelian gauge group fa tor, open string om- U(1) pa ti(cid:28) ations may ontain several anomalous 's whose anomalies are ompensated for by a generalized Green-S hwarzme hanism [15℄. In this ase, the Green-S hwarzanomaly onditions are less onstraining than U(1) in the heteroti ase, and the s aleof breakingof the anomalous 'sdepends on twisted moduli vevs,whi h are(cid:28)xedbyunknownnonperturbativephysi s. Forthesereasons,wepreferto onsiderthe aseofananomalous U(1) of the heteroti type, with its anomalies an eled by the universal Green-S hwarzme hanism. 3 depends on the ompa ti(cid:28) ation. The Ka -Moody level of a non-abelian gauge group is an integer, while the Ka -Moody level of an abelian gauge group an be fra tional. In the following, k = k 2 3 we shall assume , as is the ase in most of (if not all of) the heteroti string ompa ti(cid:28)- U(1) U(1) U(1) X X Y ations onstru ted so far. Sin e the - - anomaly annot be ompensated for C XXY by the Green-S hwarz me hanism, its oe(cid:30) ient has to vanish by itself. A ni e feature of the Green-S hwarz me hanism is that the value of the Weinberg angle C C 1 2 at the string s ale is determined by the ratio of the anomaly oe(cid:30) ients and [16℄. Indeed, upon ombining the anomaly onditions with the gauge oupling uni(cid:28) ation relation k g2 = k g2 = = g2 1 1 2 2 ··· string valid at the string s ale, one obtains: C sinθ2 (M ) = 2 . W string C +C (6) 1 2 sin2θ = 3/8 W The anoni al value of the Weinberg angle at the GUT s ale ( ) is obtained for C /C = 5/3 k /k = 5/3 1 2 1 2 , or equivalently for the normalization , whi h is well known from k = k 2 3 string model builders. As observed in Refs. [4, 5, 8℄, this normalization (together with ) mdmsmb ǫhu+hd leads through Eq. (4) to memµmτ ∼ , whi h given the un ertainty due to the order one h +h = 0, 1 2 h +h 4 u d u d oe(cid:30) ients is onsistent with data for ± or even ± . Su h values of are µ suitableforele troweak symmetry breakingifthe -termisgenerated fromthe Giudi e-Masiero me hanism [17℄. In the following, we shall (cid:28)x the Weinberg angle at its anoni al GUT value C /C = 5/3 1 2 and impose . U(1) X Another ni e feature of an anomalous symmetry is that its breaking s ale, hen e ǫ the small expansion parameter , is determined by the value of the gauge oupling at the C X = 0 2 uni(cid:28) ation s ale and by the anomaly oe(cid:30) ient [14℄. Indeed, due to the fa t that Tr 6 , a Fayet-Iliopoulos term is generated at the one-loop level [18, 19℄: g2 ξ2 = string (XM2) , 192π2 Tr (7) g M X X < 0 string Φ where is the string oupling and the redu ed Plan k mass. Provided that Tr , φ Φˆ this triggers a nonzero VEV of the s alar omponent of the super(cid:28)eld , whi h sets the value ǫ of the parameter to [14℄: g2 α string 2 ǫ = X = C , s192π2 Tr 4π 2 (8) r k g2 = g2 C /k = X/12 X = 1 where we have used the relations 2 2 string and 2 2 Tr (re all that Φ − ). As ǫ λ 0.22 we shall see later, in realisti models the value of is very lose to the Cabibbo angle, ≃ . We now look forsolutionsto the anomaly onstraints. In addition,we require the absen e of U(1) (XY) = 0 kineti mixing between the anomalous and the hyper harge, i.e. Tr . As noti ed in Ref. [20℄, this ondition also prevents the generation of a large one-loop Fayet-Iliopoulos term for the hyper harge [21℄ in the s enario in whi h the dominant ontribution to soft s alar 4 Assumingapproximatebottom-tauYukawa ouplinguni(cid:28) ationattheGUTs ale,thismassratioisbetween tanβ 5 60 h +h =0 u d 0.75 and 0.9 for the entral values of the fermion masses in the range − , whi h favours . 4 D masses omes from the anomalous -term. Su h a Fayet-Iliopoulos term ould have indu ed X harge and olour breaking minima. We lookfor models with the - harges of allquark, lepton andHiggssuper(cid:28)elds positive orzero, whi h issu(cid:30) ient toensure theuniqueness of theva uum U(1) X that breaks the anomalous (see Ref. [22℄ for a detailed dis ussion of (cid:29)at dire tions). The relevant anomaly oe(cid:30) ients read: 1 ¯ C = (q +8u¯ +2d +3l +6e¯ )+h +h , 1 A A A A A u d 3 (9) A X C = (3q +l )+h +h , 2 A A u d (10) A X ¯ C = (2q +u¯ +d ) , 3 A A A (11) A X C = (q2 2u¯2 +d¯2 l2 +e¯2)+h2 h2 , XXY A − A A − A A u − d (12) A X 2 C = (6q3 +3u¯3 +3d¯3 +2l3 +e¯3 +n3)+h3 +h3 1+C′ , X 3 A A A A A A u d − X (13) A X X = (6q +3u¯ +3d¯ +2l +e¯ +n¯ )+2(h +h ) 1+C′ , Tr A A A A A A u d − g (14) A X C′ C′ where X and g stand for the ontributions of additional SM singlets or ve tor-like represen- U(1) X tations harged under (whi h may be harged under some hidden gauge group) to the U(1) (XY) X ubi anomaly and to the mixed gravitational anomaly, respe tively. Finally, Tr reads: ¯ (XY) = 2 (q 2u¯ +d l +e¯ )+2(h h ) . A A A A A u d Tr − − − (15) A X 5 The onstraints : 3 C = C , C = C , C = 0 (XY) = 0 , 2 3 2 1 XXY 5 and Tr (16) an be rewritten as: 1 (u¯ q ) h = 0 , A A u − − 2 (17) A X 1 ¯ (d l ) h h = 0 , A A u d − − 2 − (18) A X (u¯ e¯ ) h = 0 , A A u − − (19) A X (q2 2u¯2 +d¯2 l2 +e¯2)+h2 h2 = 0 . A − A A − A A u − d (20) A X C′ C′ k 5We assume that X, g and the Ka -Moody level X are su h that the Green-S hwarz onditions are U(1) X satis(cid:28)ed for the ubi anomaly and for the mixed gravitational anomaly. 5 We look forsolutionsto Eqs. (17)-(20) whi h su essfully reprodu e the observed quark and lepton mass and mixing hierar hies with all matter harges non-negative. Sin e the Yukawa U(1) X ouplings are generated at the s ale of breaking of the anomalous (whi h for de(cid:28)niteness M 2 1016 GUT we identify with the s ale at whi h gauge ouplings unify in the MSSM, ≃ × GeV), we must onsider the masses and mixings renormalized also at this s ale. In the quark se tor, the renormalization of the CKM matrix a(cid:27)e ts, to a very good approximation, only the A λ ρ η parameter of the Wolfenstein parametrization, while , and almost do not evolve with m /m m /m d s u c energy [23, 24, 25, 26℄. The s ale dependen e of the quark mass ratios and is tanβ 3.5 50 also very weak. Numeri ally for values in the range − we (cid:28)nd: m (µ) m (M ) m (µ) m (M ) s = χ(µ) s Z , c = χ3(µ) c Z , m (µ) m (M ) m (µ) m (M ) (21) b b Z t t Z χ(M ) (0.75 0.9) GUT with ≃ − . This leads to the following approximate hierar hies of quark mixingangles and mass ratios lose to the GUT s ale, expressed inpowers of the Cabibbo angle λ 0.22 ≃ : V V λ , V λ4 , V λ2 , V λ4 λ3 , V λ2 , us cd ub cb td ts | | ≃ | | ≃ | | ∼ | | ∼ | | ∼ − | | ∼ (22) m m 1 m m d λ2 , s λ2 , u λ4 , c λ4 . m ∼ m ∼ 2 m ∼ m ∼ (23) s b c t Forlightquarkswehaveusedherethe entralvaluesofthePDGestimates[27℄. Thedependen e tanβ on of the GUT s ale values of these mass ratios and CKM elements is weak in the tanβ onsidered range of values. In the lepton se tor, the harged lepton mass hierar hy is: m 1 m e λ3 , µ λ2 . m ∼ 2 m ∼ (24) µ τ m ∆m2 m ∆m2 If one assumes a hierar hi alneutrino mass spe trum, i.e. ν3 ≈ atm and ν2 ≈ sol, the s ale dependen e of lepton mixing angles and neutrino maspses is very weak [28,p26℄, and one an take, at the s ale where right-handed neutrinos de ouple: m m U 1 , (α,A) = (e,3) , U < λ , ν2 λ , ν1 < 1 . αA e3 | | ∼ 6 | | m ∼ m (25) ν3 ν2 X Eqs. (17) to (20), together with the assumption of positive - harges for the MSSM super(cid:28)elds, are very restri tive and only a limited number of harge assignments ompatible with these onstraints give a satisfa tory a ount of the observed mass and mixing hierar hies. Let us (cid:28)rst onsider the quark se tor. The large top quark Yukawa oupling is a ounted for by q = u¯ = h = 0 3 3 u hoosing . Furthermore, the quark mass ratios and mixing angles are given by the simple expressions: muA ǫ(qA−qB)+(u¯A−u¯B) , mdA ǫ(qA−qB)+(d¯A−d¯B) , mt tanβ ǫ−(d¯3+hd) , muB ∼ mdB ∼ mb ∼ V ǫ|qA−qB| . AB ∼ (26) 6 The symbol ∼ reminds us that the above relations ontain unknown fa tors of order one, so that the a tual values of the mass ratios and mixing angles may slightly depart from the naive ǫ λ (cid:16)power ounting(cid:17). With this remark in mind, and assuming ≃ , we (cid:28)nd that the CKM q = (3,2,0) q = (4,3,0) q = (4,2,0) A A A matrix is orre tly reprodu ed by , or by . In ea h ase, λ the naive (cid:16)power ounting(cid:17) disagrees by at most one power of with the measured value of V q = (3,2,0) V V q = (4,3,0) V V ub A cb ts A us cd one or two CKM angles: | | for , | | and | | for , | | and | | q = (4,2,0) A for . In the lepton se tor, the experimental data on neutrino os illations strongly l A onstrain the (for a re ent review, see e.g. Ref. [29℄). In the absen e of holomorphi zeroes in the Dira and Majorana mass matri es, the seesaw me hanism yields an e(cid:27)e tive light neutrino mass matrix of the form: ǫ2(l1−l3) ǫ(l1−l3)+(l2−l3) ǫ(l1−l3) v2ǫ2(l3+hu) M u ǫ(l1−l3)+(l2−l3) ǫ2(l2−l3) ǫ(l2−l3) , ν ∼ M   (27) R ǫ(l1−l3) ǫ(l2−l3) 1   M R where isthes aleofright-handedneutrinomasses. Su h amassmatrix aneasilyreprodu e the hierar hi al neutrino mass spe trum. In order to reprodu e both the large atmospheri mixing angle and the hierar hy between the atmospheri and solar mass s ales, one must l = l 2 2 2 3 hoose Mand allow for a m∆imld2t/u∆nimng2betwe0e.n2 the order one entries in the lower right × submatrix of ν, of order sol atm ≈ . Due to this tuning, the solar mixing angle θ l l = 1 2 12 1 3 omes out (cid:16)large(cid:17) providped that − or (see Appendix B for details). This in turn U ǫl1−l3 e3 implies that | | ∼ should be rather large, and even lose to its present experimental l l = 1 1 3 limit in the ase − . q l A A Foragiven hoi eofthe andthe di tatedbytheCKMandPMNS mixingmatri es,the X remaining - hargesof themodelare onstrained bothby the quark and harged leptonmasses X and by Eqs. (17) to (20). We list below the solutions for the - harge assignments satisfying the onstraints (17) to (20) for whi h the predi tions obtained from naive power ounting are in reasonably good agreement with experimental data on fermion masses and mixings, when extrapolated to the GUT s ale: ¯ q = u¯ = e¯ = (3,2,0), d = l = (l ,l ,l ), h = h = 0, A A A A A 1 3 3 u d 1: [l l = 1 2; l = 0,1,2 3] 1 3 3 − or or (28) ¯ q = u¯ = e¯ = (3,2,0), d = (2,1,0), l = (1,0,0), h = 0, h = 2, A A A A A u d 2: (29) ¯ q = u¯ = e¯ = (3,2,0), d = (1,1,0), l = (1,0,0), h = 0, h = 1, A A A A A u d 3: (30) ¯ q = u¯ = e¯ = (3,2,0), d = (2,1,0), l = (2,0,0), h = 0, h = 1, A A A A A u d 4: (31) ¯ q = u¯ = e = (4,2,0), d = l = (l +1,l ,l ), h = h = 0, A A A A A 3 3 3 u d 5: [l = 0,1,2 3] 3 or (32) ¯ q = u¯ = e¯ = (4,2,0), d = (1,1,0), l = (1,0,0), h = 0, h = 1. A A A A A u d 6: (33) Several of these solutions an be found in the literature. The onstraints (17) to (20) are h = h = 0 SU(5) u d automati ally satis(cid:28)ed if and the horizontal symmetry ommutes with , as already noti ed in Ref. [20℄. This is the ase for solutions 1 and 5. We were not able to (cid:28)nd h = h = 0 SU(5) u d solutions with that are not ompatible with the symmetry. In all solutions, 7 the naive predi tions slightly depart from the observed values for some quantities, and one has to rely on the e(cid:27)e t of the un onstrained order one parameters to orre t them. In parti ular, m /m λ2 V λ3 u c ub solutions 1 to 4 naively predi t ∼ and | | ∼ , while solutions 5 and 6 predi t V V λ2 m /m l l = 1 us cd u c 1 3 | | ∼ | | ∼ , but give the orre t ratio. In addition, solution 1 with − and m /m λ2 l l = 2 m /m λ3 e µ 1 3 d s solutions 2, 3 yield ∼ ; solution 1 with − and solution 5 yield ∼ , m /m λ l l = 1 l l = 2 d s 1 3 1 3 and solution 3 yields ∼ . Finally, − is preferred over − by the solar mixing angle [30℄. l = 2 tanβ tanβ . 15 3 Solutions 1 and 5 with and solution 2 predi t a low value of , ; l = 1 tanβ (15 50) 3 solutions 1 and 5 with and solutions 3, 4 and 6 require ∼ − ; and solutions l = 0 tanβ & 50 tanβ 3 1 and 5 with require . The range of values ompatible with a given solution is rather broad due to the e(cid:27)e t of the order one oe(cid:30) ients. ǫ UsingEq. (8), one an omputethepredi tedvalueoftheexpansionparameter . Assuming α (M ) = α = 1 ( αU )1/2 2 string U 24, one obtains (the quoted values s ale as 1/24 ): ǫ = 0.23, 0.25, 0.27 l l = 1 l = 0,1,2, 1 3 3 for solution 1 with − and (34) ǫ = 0.24, 0.26, 0.28 l l = 2 l = 0,1,2, 1 3 3 for solution 1 with − and (35) ǫ = 0.24 , for solution 2, 3 and 4 (36) ǫ = 0.25, 0.27, 0.29 l = 0,1,2, 3 for solution 5 with (37) ǫ = 0.26 . for solution 6 (38) As already stressed in Ref. [14℄, it is a remarkable su ess of (cid:29)avour models based on an U(1) ǫ anomalous that the predi ted value of omes out so los¯e to the Cabibbo angle. Note q = u¯ = e¯ = (2,1,0) l = d = (1,0,0) + l A A A A A 3 that the harge assignment , , sometimes ǫ = (0.18 0.23) onsidered in the literature, predi ts − , whi h is too large for this harge assignment to be ompatible with the observed fermion masses and mixings. 3 Pre ision des ription of fermion masses and mixings The fa t that only a small number of horizontal harge assignments are phenomenologi ally a eptable, together with the interesting theoreti al aspe ts dis ussed in the previous se tion, U(1) makes the simplestanomalous modelsforfermionmasses worth further, morequantitative study. Su h a study may also be useful for studying various aspe ts of the supersymmetri (cid:29)avour problem and CP violation with hierar hi al fermion mass matri es. The harge assignments 1-6 in Eqs. (28)-(33) predi t the hierar hi al stru ture of the quark CAB and harged lepton Yukawa ouplings up to order one fa tors u,d,e, whi h should be viewed as (cid:28)xed by some unknown physi s whi h has been integrated out. The freedom in these fa tors 6 an be used to obtaina pre ise des riptionof the fermionmasses and mixings ; itis the purpose CAB of this se tion to (cid:28)nd su h sets of oe(cid:30) ients u,d,e and to dis uss their properties. 6 Foraquantitativestudyof the predi tions in the neutrinose tor,werefer to Ref. [30℄, whi h howeveronly l1 l3 = 1 onsidered the hoi e − ((cid:16)semi-anar hi al(cid:17) ase). We argued, on the basis of the analyti al formulae l1 l3 =2 given in appendix B, that the hoi e − also leads to a satisfa tory des ription of neutrino masses and 8 U(1) X Before doing so, let us noti e that, among the a eptable harge assignments, two SU(5) U(1) X (namely harge assignments 1 and 5) are onsistent with × symmetry, and an therefore be re on iled with Grand Uni(cid:28) ation of elementary for es. Moreover, those are the h = h = 0 u d only a eptable harge assignments for , for whi h, as mentioned earlier, there is some (although not strong) phenomenologi al preferen e. Charge assignments 3 and 6, SU(5) although not ompatible with symmetry, are interesting in the ontext of the supersymmetri (cid:29)avour problem be ause the right-handed down and strange quark super(cid:28)elds have the same horizontal harge, whi h in some supersymmetry breaking s enarios may suppress the squark ontribution to kaon mixing. SU(5) If one interprets the fa t that the harge assignments 1 and 5 are ompatible with × U(1) X symmetry as the manifestation of an underlying Grand Uni(cid:28)ed Theory, one should CAB = CBA u u impose the following (GUT-s ale) onstraints on the order one oe(cid:30) ients: and CAB = CBA SU(5) m /m = m /m d e . The se ond onstraint leads to the well-known relations e µ d s m /m = m /m µ τ s b and , whi h are in gross disagreement with the measured fermion masses and must be orre ted [31℄. This an be done through the ontribution of renormalizable [31℄ or non-renormalizable [3U2℄(1o)perators to the YukΣaˆwa matri es. Follow7i5ng RSeUf.([53)3℄, we shall X introdu e an additional singlet super(cid:28)eld tra5¯ns1f0orHm¯iΣˆn/gMas a of , whi h has non-renormalizable ouplings to fermions of the form . The Yukawa ouplings of SU(5) U(1) X the down-type quarks and harged leptons then arise from the two × invariant superpotential terms: Σˆ Φˆ d¯A+qB+hd W = 5¯ACAB10BH¯ + 5¯ACAB10BH¯ , 1 M 2 ! M! (39) Φˆ Σˆ whi h, after the s alar omponents of and a quire VEVs, lead to: YAB = CAB +κCAB ǫd¯A+qB+hd, d 1 2 YeAB = (cid:0)C1BA −3κC2BA(cid:1) ǫd¯B+qA+hd, (40) (cid:0) (cid:1) κ Σ /M κ = 0.3 where ≡ h i . In our numeri al (cid:28)ts, we take , whi h makes it easy to a ount for the di(cid:27)eren e between down-type quark and harged letpon masses. U(1) X In order to test in a quantitative way the ability of the symmetry to des ribe the 24 CAB = CBA CAB fermion masses and mixings, we look for sets of omplex oe(cid:30) ients u u , 1 and CAB 9 4 2 that reprodu e the quark and harged lepton masses and the parameters of the CKM matrix. It is also interesting to he k how well the various harge assignments reprodu e the observed values of the fermion masses and mixings for randomly generated oe(cid:30) ients. Thi¯s is l = d A A illustrated in Figs. 1 and 2 for harge assignments 1 (with two di(cid:27)erent hoi es for ), CAB 5 and 6. For randomly generated sets of oe(cid:30) ients { u,1,2} with arbitrary phases and moduli 0.3 3 in the range − , ea h histogram shows the relative number of sets of oe(cid:30) ients leading R(Q) Q(M ) /Q(M ) GUT predicted GUT evolved to a given value of the ratio ≡ for some parti ular mixings. In the following, we shallnot dis ussthe neutrino se toragainand shall fo us on the hargedfermion se tor. 9